Haar null and Haar meager sets: a survey and new results

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.


Introduction and history
Many results in various branches of mathematics state that certain properties hold for almost every element of a space. In the continuous, large structures which are frequently studied in analysis it is common to encounter a property that is true for most points, but false on a negligibly small part of the structure. These situations mean that there are facts which can be grasped only by defining a suitable notion of smallness and stating that the exceptional elements form a small set.
In the Euclidean space R n there is a generally accepted, natural notion of smallness: a set is considered to be small if it has Lebesgue measure zero. The Haar measure (which was introduced by Alfréd Haar in 1933) generalizes the Lebesgue measure for arbitrary locally compact topological groups (see Subsection 3.3 for a brief introduction). Although 'the' Haar measure is not completely unique (except in compact groups, where there exists a natural choice), the system of sets of Haar measure zero is well defined. This yields a very useful notion of smallness in locally compact groups, but says nothing of non-locally compact groups, where there is no Haar measure.
In the paper [23] (which was published in 1972), Christensen introduced the notion of Haar null sets, which is equivalent to having Haar measure zero in locally compact groups and defined in every abelian Polish group. (A topological group is Polish if it is separable and completely metrizable, for the definition of Haar nullness see Definition 3.1.1.) Since then lots of papers were published which either study some property of Haar null sets or use this notion of smallness to state facts which are true for almost every element of some structure. It was relatively easy to generalize this notion to nonabelian groups, on the other hand, the assumption that the topology is Polish is still usually assumed, because it turned out to be convenient and useful.
There is another widely used notion of smallness, the notion of meager sets (also known as sets of the first category). Meager sets can be defined in any topological space; a set is said to be meager if it is the countable union of nowhere dense sets (that is, sets which are not dense in any open set). A topological space is called a Baire space if the nonempty open sets are nonmeager; this basically means that one can consider the meager sets small in these spaces. The Baire category theorem states that all completely metrizable spaces and all locally compact Hausdorff spaces are Baire spaces (see [67,Theorem 8.4] for the proofs).
In locally compact groups the system of meager sets and the system of sets of Haar measure zero share many properties. For example the classical Erdős-Sierpiński duality theorem states that it is consistent that there is a bijection f : R → R such that for every set A ⊆ R, f (A) is meager if and only if A has Lebesgue measure zero and f (A) has Lebesgue measure zero if and only if A is meager. Despite this, there are sets that are small in one sense and far from being small in the other sense, for example every locally compact group can be written as the union of a meager set and a set of Haar measure zero.
In 2013, Darji defined the notion of Haar meager sets in the paper [29] to provide a better analog of Haar null sets in the non-locally compact case. (For the definition see Definition 3.1.6.) Darji only considered abelian Polish groups, but [38] generalized this notion to nonabelian Polish groups. Haar meager sets coincide with meager sets in locally compact Polish groups, and Haar meagerness is a strictly stronger notion than meagerness in non-locally compact abelian groups (see Theorems 3.3.13 and 3.3.14). The analogy between Haar nullness and Haar meagerness means that most of the results for Haar null sets are also true for Haar meager sets and it is often possible to prove them using similar methods.
First in Section 2, we introduce the notions, definitions, and conventions which are not related to our area, but used repeatedly in this paper. Then Section 3 defines the core notions and investigates their most important properties.
After these, Section 4 considers the modified variants of the definitions. This section starts with a large collection of equivalent definitions for our core notions, then lists and briefly describes most of the versions which appear in the literature and are not (yet proved to be) equivalent to the 'plain' versions.
The next section, Section 5, considers the feasibility of generalizing four well-known results (Fubini's theorem, the Steinhaus theorem, the countable chain condition, writing the whole space as the union of a null and a meager set) for non-locally compact Polish groups.
Then in Section 6 we discuss some proof techniques for questions from this area. Some of these are essentially useful lemmas, the others are just ideas and ways of thinking which can be helpful in certain cases.
Finally, in Section 7 we give a highly incomplete list of applications of Haar null sets in various fields of mathematics and in Section 8 we collect the open questions that are mentioned in this survey.

Notation and terminology
This section is the collection of the miscellaneous notations, definitions, and conventions that are used repeatedly in this paper.
The symbols N and ω both refer to the set of nonnegative integers. We write N if we consider this set as a topological space (with the discrete topology) and ω if we use it only as a cardinal, ordinal, or index set. (For example, we write the Polish space of the countably infinite sequences of natural numbers as N ω .) We consider the nonnegative integers as von Neumann ordinals, that is, we identify the nonnegative integer n with the set {0, 1, 2, . . . , n − 1}. P(S) denotes the power set of a set S. For a set S ⊆ X × Y , x ∈ X, and y ∈ Y , S x is the x-section S x = {y : (x, y) ∈ S} and S y is the y-section S y = {x : (x, y) ∈ S}.
If S is a subset of a topological space, int(S) is the interior of S and S is the closure of S. We consider N, Z, and all finite sets to be topological spaces with the discrete topology. (Note that this convention allows us to simply write the Cantor set as 2 ω = {0, 1} ω .) If X is a topological space, then B(X) denotes its Borel subsets (B(X) is the σ-algebra generated by the open sets, see [67, Chapter II]), M(X) denotes its meager subsets (a set is meager if it is the union of countably many nowhere dense sets and a set is nowhere dense if the interior of its closure is empty, see [67, § 8.A]).
If the space X is Polish (that is, separable and completely metrizable), then Σ 1 1 (X) denotes its analytic subsets (a set is analytic if it is the continuous image of a Borel set, see [67, Chapter III]), Π 1 1 (X) denotes its coanalytic subsets (a set is coanalytic if its complement is analytic, see [67,Chapter IV]).
If the topological space X is clear from the context, we simply write B, M, Σ 1 1 and Π 1 1 . In a metric space (X, d), diam(S) = sup{d(x, y) : x, y ∈ S} denotes the diameter of the subset S. If x ∈ X and r > 0, then B(x, r) = {x ∈ X : d(x, x ) < r} and B(x, r) = {x ∈ X : d(x, x ) r} denotes, respectively, the open and the closed ball with center x and radius r in X.
If μ is an outer measure on a set X, we say that A ⊆ X is μ-measurable if μ(B) = μ(B ∩ A) + μ(B \ A) for every B ⊆ X. Unless otherwise stated, we identify an outer measure μ with its restriction to the μ-measurable sets and 'measure' means an outer measure or the corresponding complete measure. A measure μ is said to be Borel if all Borel sets are μ-measurable. The support of the measure μ is denoted by supp μ.
Almost all of our results will be about topological groups. A set G is called a topological group if is equipped with both a group structure and a Hausdorff topology and these structures are compatible, that is, the multiplication map G × G → G, (g, h) → gh and the inversion map G → G, g → g −1 are continuous functions. We use the convention that whenever we require a group to have some topological property (for example, a 'compact group', a 'Polish group', . . .), then it means that the group must be a topological group and have that property (as a topological space). The identity element of a group G will be denoted by 1 G .
Most of the results in this paper are about certain subsets of Polish groups. Unless otherwise noted, (G, ·) denotes an arbitrary Polish group. We denote the group operation by multiplication even when we assume that the (abstract) group under consideration is abelian, but we write the group operation of well-known concrete abelian groups like (R, +) or (Z ω , +) as addition.
(1) The system HN of Haar null sets is a σ-ideal. (2) The system GHN of generalized Haar null sets is a σ-ideal.
Proof. It is trivial that both HN and GHN satisfy (I) and (II) in Definition 3.2.1. The proof of (III) that is reproduced here is from the Appendix of [26], where a corrected version of the proof in [75] is given. Proving this fact is easier in abelian Polish groups (see [23,Theorem 1]) and when the group is metrizable with a complete left invariant metric (this would allow the proof of [75,Theorem 3] to work without modifications). The Appendix of [26] mentions the other approaches and discusses the differences between them.
The proof of (III) for Haar null and for generalized Haar null sets is very similar. The following proof will be for Haar null sets, but if 'Borel set' is replaced with 'universally measurable set' and 'Haar null' is replaced with 'generalized Haar null', it becomes the proof for generalized Haar null sets.
Let A n be Haar null for all n ∈ ω. By definition there are Borel sets B n ⊆ G and Borel probability measures μ n on G such that A n ⊆ B n and μ n (gB n h) = 0 for every g, h ∈ G. Let d be a complete metric on G that is compatible with the topology of G (as G is Polish, it is completely metrizable).
We construct for all n ∈ ω a compact set C n ⊆ G and a Borel probability measureμ n such that the support ofμ n is C n ,μ n (gB n h) = 0 for every g, h ∈ G (that is,μ n is a witness measure) and the 'size' of the sets C n decreases 'quickly'.
The construction will be recursive. For the initial step use Corollary 3.2.3 to find a Borel probability measureμ 0 that satisfiesμ 0 (gB 0 h) = 0 for every g, h ∈ G and that has compact support C 0 ⊆ G. Assume thatμ n and C n are already defined for all n < n. By Lemma 3.2.4, there exists a neighborhood U n of 1 G such that if u ∈ U n , then d(k · u, k) < 2 −n for every k in the compact set C 0 C 1 C 2 · · · C n−1 . Applying Corollary 3.2.3 again we can find a Borel probability measureμ n that satisfiesμ n (gB n h) = 0 for every g, h ∈ G and that has a compact support C n ⊆ U n .
If c n ∈ C n for all n ∈ ω, then it is clear that the sequence (c 0 c 1 c 2 · · · c n ) n∈ω is a Cauchy sequence. As (G, d) is complete, this Cauchy sequence is convergent; we write its limit as the infinite product c 0 c 1 c 2 · · · . The map ϕ : n∈ω C n → G, ϕ((c 0 , c 1 , c 2 , . . .)) = c 0 c 1 c 2 · · · is the uniform limit of continuous functions, hence it is continuous.
We claim that μ witnesses that A = n∈ω A n is Haar null. Note that A is contained in the Borel set B = n∈ω B n , so it is enough to show that μ(gBh) = 0 for every g, h ∈ G. As μ is σ-additive, it is enough to show that μ(gB n h) = 0 for every g, h ∈ G and n ∈ ω.
By the definition of μ this means that μ(gB n h) = 0.
Proof. Again, the proof of (I) and (II) in Definition 3.2.1 is obvious. We reproduce the proof of (III) from [38,Theorem 3]. This proof will be very similar to the proof of Theorem 3.2.5, but restricting the witnesses to a smaller 'part' of G is simpler in this case (we do not need an analogue of Corollary 3.2.3).
Let A n be Haar meager for all n ∈ ω. By definition there are Borel sets B n ⊆ G, compact metric spaces K n = ∅, and continuous functions f n : K n → G such that f −1 n (gB n h) is meager in K n for every g, h ∈ G. Let d be a complete metric on G that is compatible with the topology of G.
We construct for all n ∈ ω a compact metric spaceK n and a continuous functionf n :K n → G satisfying thatf −1 n (gB n h) is meager inK n for every g, h ∈ G (that is,f n is a witness function) and the 'size' of the imagesf n (K n ) ⊆ G decreases 'quickly'.
Unlike the Haar null case, we do not have to apply recursion in this construction. By Lemma 3.2.4 there exists a neighborhood U n of 1 G such that if u ∈ U n , then d(k · u, k) < 2 −n for every k in the compact set f 0 (K 0 )f 1 (K 1 ) · · · f n−1 (K n−1 ). Let x n ∈ f n (K n ) be an arbitrary element andK n = f −1 n (x n U n ). The setK n is compact (because it is a closed subset of a compact set) and nonempty. Letf n :K n → G,f n (k) = x −1 n f n (k), this is clearly continuous.
Proof. Fix n ∈ ω and g, h ∈ G. The setf −1 n (U n ) is open in K n and because f is a witness function, the setf −1 n (gB n h) = f −1 n (x n gB n h) is meager in K n . This means thatf −1 Since each open subset of K n is comeager in its closure and the closure off −1 Let K be the compact set n∈ωK n and for n ∈ ω let ψ n be the continuous function ψ n : By the choice of U n we obtain d(ψ n−1 (k), ψ n (k)) 2 −n for every k ∈ K. Using the completeness of d this means that the sequence of functions (ψ n ) n∈ω is uniformly convergent. Let f : K → G be the limit of this sequence. f is continuous, because it is the uniform limit of continuous functions.
We claim that f witnesses that A = n∈ω A n is Haar meager. Note that A is contained in the Borel set B = n∈ω B n , so it is enough to show that f −1 (gBh) is meager in K for every g, h ∈ G. As meager subsets of K form a σ-ideal, it is enough to show that f −1 (gB n h) is meager in K for every g, h ∈ G and n ∈ ω.
Fix g, h ∈ G and n ∈ ω. Note that if k j ∈K j for every j = n, j ∈ ω, then Claim 3.2.7 means that {k n ∈K n : f (k 0 , k 1 , . . . , k n , . . .) ∈ gB n h} is meager inK n . Applying the Kuratowski-Ulam theorem (see, for example, [67,Theorem 8.41]) in the product space ( j =nK j ) ×K n , the Borel set f −1 (gB n h) is meager.
As the group G acts on itself via multiplication, it is useful if this action does not convert 'small' sets into 'large' ones. This means that a 'nice' notion of smallness must be a translation invariant system. Proof. If x ∈ G, then any nonempty set in I has a translate that contains {x} as a subset, hence {x} ∈ I. Using that I is closed under countable unions, this yields that if A ⊆ G is countable, then A ∈ I. To prove the other claim, suppose for a contradiction that U ⊆ G is a nonempty open set that is in I. It is clear that G = g∈G gU , and as G is Lindelöf, G = n∈ω g n U for some countable subset {g n : n ∈ ω} ⊆ G. But here g n U ∈ I (because I is translation invariant) and thus G ∈ I (because I is closed under countable unions), and this means that I contains all subsets of G, and this is a contradiction.
Remark 3.2.11. Let I be one of the σ-ideals HN , GHN , and HM. If G is countable, then I = {∅}, otherwise I contains a nonempty set and does not contain all subsets of G.

Connections to Haar measure and meagerness
This subsection discusses the connection between sets with Haar measure zero and Haar null sets and the connection between meager sets and Haar meager sets. In the simple case when G is locally compact, we will find that equivalence holds for both pairs, justifying the names 'Haar null' and 'Haar meager'. When G is non-locally compact, we will see that the first connection is broken by the fact that there is no Haar measure on the group. For the other pair we will see that Haar meager sets are always meager, but there are Polish groups where the converse is not true.
First we recall some well-known facts about Haar measures. For proofs and more detailed discussion, see, for example, [58, § 15].
If left invariance is replaced by the property λ(Bg) = λ(B) for all B ∈ B(G) and g ∈ G (right invariance), the measure is called a right Haar measure. If G is compact, (I) means that the left and right Haar measures are finite measures, and this fact can be used to prove the following result: This result is also trivially true in abelian locally compact groups, but not true in all locally compact groups. However, the following result remains true: This allows us to define the following notion: Definition 3.3.6. Suppose that G is a locally compact group and fix an arbitrary left (or right) Haar measure λ. We say that a set N ⊆ G has Haar measure zero if N ⊆ B for some Borel set B with λ(B) = 0. The collection of these sets is denoted by N = N (G). Definition 3.3.2 defines the Haar measures only on the Borel sets. If λ is an arbitrary left (or right) Haar measure, we can complete it using the standard techniques. The domain of the completion will be σ(B(G) ∪ N ) (the σ-algebra generated by N and the Borel sets). For every set A in this σ-algebra, let This completion will be a complete measure that agrees with the original λ on Borel sets and satisfies properties (I)-(IV) from Definition 3.3.2 (or right invariance instead of left invariance if λ was a right Haar measure). We will identify a left (or right) Haar measure with its completion and we will also call this extension (slightly imprecisely) a left (or right) Haar measure.
The following theorem justifies the choice of the name of 'Haar null' sets. The foundational paper [23] shows this in the abelien case; [75, Theorem 1] gives a complete proof (using a slightly different definition of Haar measures). Theorem 3.3.7 (Christensen, Mycielski). If G is a locally compact Polish group, then the system of sets with Haar measure zero is the same as the system of Haar null sets and is the same as the system of generalized Haar null sets, that is, N (G) = HN (G) = GHN (G).
Proof. N (G) ⊆ HN (G): Let λ be a left Haar measure and λ be a right Haar measure. If N ∈ N (G) is arbitrary, then by definition there is a Borel set B satisfying N ⊆ B and λ(B) = 0. The left invariance of G means that λ(gB) = 0 for every g ∈ G. Applying Theorem 3.3.5 this means that λ (gB) = 0 for every g ∈ G, and applying the right invariance of λ we get that λ (gBh) = 0 for every g, h ∈ G. Using the regularity of λ , it is easy to see that there is a compact set is clearly a Borel probability measure. μ λ means that μ(gBh) = 0 for every g, h ∈ G, so B and μ satisfy the requirements of Definition 3.1.1.
HN (G) ⊆ GHN (G): This is obviously true in all Polish groups, see Remark 3.1.5.
By definition there exists a universally measurable B ⊆ G and a Borel probability measure μ such that μ(gBh) = 0 for every g, h ∈ G. Note that we will only use that μ(Bh) = 0 for every h ∈ G, so we will also prove that (using the terminology of Subsection 4.4) all generalized right Haar null sets have Haar measure zero. Let λ be a left Haar measure on G. Let m be the multiplication map m : Note that the set m −1 (B) = {(x, y) ∈ G × G : x · y ∈ B} is universally measurable in G × G, because it is the preimage of a universally measurable set under the continuous map m. (This follows from that the preimage of a Borel set under m is Borel, and for every σ-finite measure ν on G × G, the preimage of a set of m * (ν)-measure zero under m must be of ν-measure zero.
Here m * (ν) is the push-forward measure: Applying Fubini's theorem in the product space G × G to the product measure μ × λ (which is a σ-finite Borel measure), we get that As μ is a witness measure, the right-hand side is the integral of the constant 0 function. On the left-hand side λ(x −1 B) = λ(B), as λ is left invariant (note that B is λ-measurable, because B is universally measurable and λ is σ-finite). Thus 0 = G λ(B) d μ(x) = λ(B). As A ⊆ B, this means that λ(A) = 0, A ∈ N (G).
We reproduce the proof of the classical theorem which shows that (left and right) Haar measures do not exist on topological groups that are not locally compact. We will apply the following generalized version of the Steinhaus theorem. (The original version by Steinhaus considered the Lebesgue measure on the group (R, +), Weil generalized this for the case of Haar measures [88, p. 50].) Proof. As λ is a Haar measure and C is compact, λ(C) < ∞. Using the regularity of λ, there is an open set U ⊇ C that satisfies λ(U ) < 2λ(C).
Proof. For every c ∈ C the multiplication map m : Then v · C and C are subsets of U and λ(v · C) = λ(C) > λ(U ) 2 (we used the left invariance of λ). This means that v · C ∩ C = ∅, so there exists c 1 , c 2 ∈ C with vc 1 = c 2 , but this means that v = c 2 c −1 We note that in Polish groups it is possible to find a compact subset with positive Haar measure in every set with positive Haar measure. Hence the following version of the previous theorem is also true: Other, more general variants of the Steinhaus theorem are examined in Subsection 5.2. Now we are ready to prove that Haar measures only exist in the locally compact case: Theorem 3.3.11. If G is a topological group and λ is a left Haar measure on G, then G is locally compact.
Proof. λ(G) > 0 as G is open. Using the regularity of G, there exists a compact set C with λ(C) > 0. The set C · C −1 is compact (it is the image of the compact set C × C under the continuous map (x, y) → xy −1 ). Applying Theorem 3.3.8 yields that C · C −1 is a neighborhood of 1 G , but then for every g ∈ G the set g · C · C −1 is a compact neighborhood of g, and this shows that G is locally compact.
The connection between meager sets and Haar meager sets is simpler. The following results are from [29] (the first paper about Haar meager sets, which only considers abelian Polish groups) and [38] (where the concept of Haar meager sets is extended to all Polish groups). Proof. Let A be a Haar meager subset of G. By definition there exists a Borel set B ⊇ A, a (nonempty) compact metric space K, and a continuous function f : K → G such that f −1 (gBh) is meager in K for every g, h ∈ G.
Consider the set which is Borel because it is the preimage of B under the continuous map (g, k) → g −1 · f (k). For every g ∈ G, the g-section of this set is S g = {k ∈ K : f (k) ∈ gB} = f −1 (gB), and this is a meager set in K. Hence, by the Kuratowski-Ulam theorem, S is meager in G × K. Using the Kuratowski-Ulam theorem again, for comeager many k ∈ K, the section Since K is compact, there is at least one such k. Then the inverse of the homeomorphism b → f (k) · b −1 maps the meager set S k to B, and this shows that B is meager. Proof. We only need to prove the inclusion M(G) ⊆ HM(G). As G is locally compact, there is a nonempty open set U ⊆ G such that U is compact. Let f : U → G be the identity map restricted to U . If M is meager in G, then there exists a meager Borel set B ⊇ M . The set gBh is meager in G for every g, h ∈ G (as x → gxh is a homeomorphism), so f −1 (gBh) = gBh ∩ U is meager in U for every g, h ∈ G. Theorem 3.3.14 (Darji, Doležal-Rmoutil-Vejnar-Vlasák). In a non-locally compact Polish group G that admits a two-sided invariant metric meagerness is a strictly stronger notion than Haar meagerness, that is, HM(G) M(G).
Proof. We know that HM(G) ⊆ M(G). To construct a meager but not Haar meager set, we will use a theorem from [83]. As the proof of this purely topological theorem is relatively long, we do not reproduce it here. Theorem 3.3.15 (Solecki). Assume that G is a non-locally compact Polish group that admits a two-sided invariant metric. Then there exists a closed set F ⊆ G and a continuous function ϕ : F → 2 ω such that for any x ∈ 2 ω and any compact set Using this we construct a closed nowhere dense set M that is not Haar meager. The system {f −1 ({x}) : x ∈ 2 ω } contains continuum many pairwise disjoint closed sets. If we fix a countable basis in G, only countably many of these sets contain an open set from that basis. If for x 0 ∈ 2 ω the set M := f −1 ({x 0 }) does not contain a basic open set, then it is nowhere dense (as it is closed with empty interior). On the other hand, it is clear that M is not Haar meager, as for every compact metric space K and continuous function f :

Alternative definitions
In this section, first we discuss various alternative definitions which are equivalent to the 'normal' definitions, but may be easier to prove or easier to use in some situations. After this, we will briefly describe some other versions which appeared in papers about this topic.

Equivalent versions
In this subsection, we mention some alternative definitions which are equivalent to Definitions 3.1.1, 3.1.4 or 3.1.6. Most of these equivalences are trivial, but even these trivial equivalences can be frequently used as lemmas. First we list some versions of the definition of Haar null sets. (1) There exists a Borel set B ⊇ A and a Borel probability measure μ on G such that μ(gBh) = 0 for every g, h ∈ G (that is, A is Haar null).
(2) There exists a Borel Haar null set B ⊇ A. On the other hand, Theorem 4.2.1 states that we cannot replace 'analytic' by 'coanalytic' in (4) and (5).
Proof. First note that Lusin's theorem (see [67, 29.7]) states that all analytic sets are universally measurable, hence gBh is μ-measurable in condition (4).  (3) is true, then there exists a Borel probability measure μ such that μ(gB h) = 0 for some (universally measurable) B ⊇ B and every g, h ∈ G, but this means that μ(gBh) = 0 for every g, h ∈ G.
The implication (1) ⇒ (4) follows from the fact that all Borel sets are analytic. The implication (4) ⇒ (5) is trivial again, considering that all analytic sets are universally measurable.
Finally we prove (5) ⇒ (1) to conclude the proof of the theorem. This proof is reproduced from [83]. Without loss of generality we may assume that the set A itself is analytic generalized Haar null. We have to prove that there exists a Borel set B ⊇ A and a Borel probability measure μ on G such that μ(gBh) = 0 for every g, h ∈ G.
By definition there exists a Borel probability measure μ such that μ(gBh) = 0 for some (universally measurable)B ⊇ A and every g, h ∈ G, but this means that μ(gAh) = 0 for every g, h ∈ G. is coanalytic on analytic, that is, for every Polish space Y and P ∈ Σ 1 1 (Y × G), the set {y ∈ Y : P y ∈ Φ} is Π 1 1 .
Proof. Let Y be Polish space and P ∈ Σ 1 1 (Y × G) and let ThenP is analytic, as it is the preimage of P under (g, h, y, γ) → (y, g −1 γh −1 ). We will use the following result, which can be found as [67,Theorem 29.26]: (Standard Borel spaces are introduced in [67, Section 12], we only use the fact that every Polish space is a standard Borel space.) Using this result yields that is analytic, therefore its section at (ν = μ, r = 0), which is the set is also analytic. Projecting this on Y yields that {y ∈ Y : μ(P (g,h,y) ) > 0 for some g, h ∈ G} is analytic, but then {y ∈ Y : μ(P (g,h,y) ) = 0 for all g, h ∈ G} = {y ∈ Y : P y ∈ Φ} is coanalytic. Now, since A ∈ Φ, by the dual form of the First Reflection Theorem (see [67,Theorem 35.10] and the remarks following it), there exists a Borel set B with B ⊇ A and B ∈ Φ, and this B (together with μ) satisfies our requirements.
We note that in Subsection 4.6 we will use a modified variant of this method to prove the stronger result Theorem 4.6.11. In Definitions 3.1.1 and 3.1.4 the witness measure is required to be a Borel probability measure, but some alternative conditions yield equivalent definitions. A set A ⊆ G is Haar null (or generalized Haar null) if and only if there is a Borel (or universally measurable) set B ⊇ A that satisfies the equivalent conditions listed in the following theorem. (1) There exists a Borel probability measure μ on G such that μ(gBh) = 0 for every g, h ∈ G, (2) There exists a Borel probability measure μ on G such that μ has compact support and μ(gBh) = 0 for every g, h ∈ G, (3) There exists a Borel measure μ on G such that 0 < μ(X) < ∞ for some μ-measurable set X ⊆ G and μ(gBh) = 0 for every g, h ∈ G, (4) There exists a Borel measure μ on G such that 0 < μ(C) < ∞ for some compact set C ⊆ G and μ(gBh) = 0 for every g, h ∈ G (the paper [62]   (1) There exists a Borel probability measure μ on G such that μ(gBh) = 0 for every g, h ∈ G. (5) There exists a Borel probability measure μ on G and a generalized Haar null set N ⊆ G such that μ(gBh) = 0 for every g, h ∈ G \ N .
Proof. The direction (1) ⇒ (5) is trivial. To prove the other direction, assume that (5) holds. We may assume without loss of generality that N is universally measurable. Fix a Borel probability measure ν on G such that ν(gN h) = 0 for all g, h ∈ G and let ν denote the Borel probability measure ν(X) = ν({x −1 : x ∈ X}).
Consider the measureμ defined by the convolutioñ which is clearly a Borel probability measure on G. To prove thatμ satisfies condition (1) we need to check that if we fix arbitrary g, h ∈ G, theñ We may apply Fubini's theorem to see that We know that ν(h −1 N ) = ν(Ng −1 ) = 0 and so the set But here p ∈ G \ Ng −1 and r ∈ G \ h −1 N means that pg, hr ∈ G \ N and therefore μ(pgBhr) = 0, concluding our proof. We remark that the proof only used the fact that N is a generalized left-and-right Haar null set. (This is a weaker notion than generalized Haar null sets, we discuss it in Subsection 4.4.) The following result characterizes Haar null sets with witness functions, analogously to Haar meager sets: (1) G is Haar null.
(2) There exists an injective continuous map f : (3) There exists a continuous map f : Here N (2 ω ) is the σ-ideal of sets of Haar measure zero on the Cantor cube 2 ω . The relatively long proof of this result can be found as [9,Theorem 4.3]. The paper [9] only considers the case of abelian Polish groups, but the proof of this result remains valid in the case when G is not necessarily abelian. For related results stated in this survey, see also Theorems 4.1.11 and 4.6.3.
The following result gives an equivalent characterization of Haar null sets which allows proving that a Borel set is Haar null by constructing measures that assign small, but not necessarily zero measures to the translates of that set. In [70, Theorem 1.1] Matoušková proves this theorem for separable Banach spaces, but her proof can be generalized to work in arbitrary Polish groups.
Theorem 4.1.7 (Matoušková). A Borel set B is Haar null if and only if for every ε > 0 and neighborhood U of 1 G , there exists a Borel probability measure μ on G such that the support of μ is contained in U and μ(gBh) < ε for every g, h ∈ G.
Proof. Let P (G) be the set of Borel probability measures on G. As G is Polish, [67,Theorem 17.23] states that P (G) (endowed with the weak topology) is also a Polish space. In particular, this means that it is possible to fix a compatible metric d such that (P (G), d) is a complete metric space. If μ, ν ∈ P (G), let (μ * ν)(X) = (μ × ν)({(x, y) : xy ∈ X}) be their convolution. It is straightforward to see that * is associative (but not commutative in general, as we did not assume that G is commutative). The map * : P (G) × P (G) → P (G) is continuous, for a proof of this see, for example, [59,Proposition 2.3]. Let δ(X) = 1 if 1 G ∈ X, and δ(X) = 0 if 1 G / ∈ X, then it is clear that δ ∈ P (G) is the identity element for * . First we prove the 'only if' part. Let (U n ) n∈ω be open sets with n U n = {1 G }. For every n ∈ ω, fix a Borel probability measure μ n such that (I) supp μ n ⊆ U n and (II) μ n (gBh) < 1 n+1 for every g, h ∈ G.
It is easy to see from property (I) that the sequence (μ n ) n∈ω (weakly) converges to δ, and this and the continuity of * means that for any ν ∈ P (G) the sequence d(ν, ν * μ n ) converges to zero. This allows us to replace (μ n ) n∈ω with a subsequence which also satisfies that d(ν, ν * μ n ) < 2 −n for every measure ν from the finite set {μ j0 * μ j1 * . . . * μ jr : r < n and 0 j 0 < j 1 < · · · < j r < n}.
We have to prove that μ(gBh) = 0 for every g, h ∈ G. To show this fix arbitrary g, h ∈ G and n ∈ ω; we will show that μ(gBh) 1 n+1 . Let α n = μ 0 * μ 1 * . . . * μ n−1 and β n = μ n+1 * μ n+2 * . . . and note the continuity of * yields This means that To prove the 'if' part of the theorem, suppose that there exists a ε > 0 and a neighborhood U of 1 G such that for every Borel probability measure μ on G if supp μ ⊆ U , then μ(gBh) ε for some g, h ∈ G. Let μ be an arbitrary Borel probability measure. Applying Lemma 3.2.2 yields that there are a compact set C ⊆ G and c ∈ G with μ(C) > 0 and C ⊆ cU . Define μ (X) = μ(cX∩C) μ(C) , then μ is a Borel probability measure with supp μ ⊆ U , hence μ (gBh) ε for some g, h ∈ G. This means that μ(cgBh) = 0, so μ is not a witness measure for B, and because μ was arbitrary, B is not Haar null. (1) There exists a Borel set B ⊇ A, a (nonempty) compact metric space K, and a continuous function f : (2) There exists a Borel Haar meager set B ⊇ A.
Without loss of generality we may assume that the set A itself is analytic and satisfies that f −1 (gAh) is meager in K for every g, h ∈ G for some (nonempty) compact metric space K and continuous function f : K → G. We will prove that (for this K and f ) there exists a Borel set ThenP is analytic, as it is the preimage of P under (g, h, y, k) → (y, g −1 f (k)h −1 ). Novikov's theorem (see, for example, [67,Theorem 29.22]) states that if U and V are Polish spaces To prove our next result we will need a technical lemma. This is a modified version of the well-known result that for every (nonempty) compact metric space K, there exists a continuous surjective map ϕ : 2 ω → K.
Proof. We will use a modified version of the usual construction. Note that if diam(K) < 1, then this function ϕ will be surjective (we will not need this fact).
Let 2 <ω = n∈ω 2 n be the set of finite 0-1 sequences. The length of a sequence s is denoted by |s| and the sequence of length zero is denoted by ∅. If s and t are sequences (where t may be infinite), s t means that s is an initial segment of t (that is, the first |s| elements of t form the sequence s) and s ≺ t means that s t and s = t. For sequences s, t where |s| is finite, s t denotes the concatenation of s and t. For a finite sequence s, let [s] = {x ∈ 2 ω : s x} be the set of infinite sequences starting with s. Note that [s] is clopen in 2 ω for every s ∈ 2 <ω .
We will choose a set of finite 0-1 sequences S ⊆ 2 <ω , and for every s ∈ S we will choose a point k s ∈ K. The construction of S will be recursive: we recursively define for every n ∈ ω a set S n and let S =˙ n∈ω S n . Our choices will satisfy the following properties.
(I) For every n ∈ ω and x ∈ 2 ω , there exists a unique s = s(x, n) ∈ S n with s x, and (II) For every n ∈ ω and x ∈ 2 ω , the set C x,n = 0 j<n B(k s(x,j) , 1 j+1 ) is nonempty.
First we let S 0 = {∅} and choose an arbitrary k ∅ ∈ K, then these trivially satisfy (I) and (II).
Suppose that we already defined S 0 , S 1 , . . . , S n−1 and let s ∈ S n−1 be arbitrary. Note that for x, x ∈ [s], C x,n−1 = C x ,n−1 and denote this common set with C s . The set C s is (nonempty) compact and it is covered by the open sets {B(c, 1 n ) : c ∈ C s }, hence we can select a collection (c We may increase the cardinality of this collection by repeating one element several times if necessary, and thus we can assume that the index set I s is of the form 2 s for some integer s 1 (that is, it consists of the 0-1 sequences with length s ). Now we can define S n = {s t : s ∈ S n−1 , t ∈ 2 s }. If s ∈ S n , then there is a unique s ∈ S n−1 such that s s , if t satisfies that s = s t (that is, t is the final segment of s ), then let k s = c It is straightforward to check that these choices satisfy (I). Property (II) is satisfied because if x ∈ 2 ω , and s, s , and t are the sequences that satisfy s = s(x, n − 1) and s t = s = s(x, n), n ), and this shows that C x,n is not empty.
We use this construction to define the function ϕ: let {ϕ(x)} = n∈ω C x,n (as the system of nonempty compact sets (C x,n ) n∈ω is descending and diam(C x,n ) diam(B(k s(x,n−1) , 1 n )) 2 · 1 n → 0, the intersection of this system is indeed a singleton). This function is continuous, because if ε > 0 and ϕ(x) = k ∈ K, then 2 · 1 n0+1 < ε for some n 0 , and then for every x in the clopen set [s(x, : s ∈ S} is a base of the topology of 2 ω . This means that there exist a n ∈ ω \ {0} and contains an open set for some n ∈ ω. But then F n contains an open set, which is a contradiction. We use this lemma to show that in Definition 3.1.6 we can also restrict the choice of the compact metric space K. In the following theorem the equivalence (1)   (1) There exists a (nonempty) compact metric space K and a continuous function f : (2) There exists a continuous function f : Proof. (1) ⇒ (2): This implication is an easy consequence of Lemma 4.1.10. If K and f satisfies the requirements of (1) and ϕ is the function granted by Lemma 4.1.10, theñ f = f • ϕ : 2 ω → G will satisfy the requirements of (2), because it is continuous and for every (2) ⇒ (3): If G is countable, the only Haar meager subset of G is the empty set. In this case, any nonempty C ⊆ G and continuous function f : C → G is sufficient. If G is not countable, then it is well known that there is a (compact) set C ⊆ G that is homeomorphic to 2 ω .
Composing the witness function f : 2 ω → G granted by (2) with this homeomorphism yields a function that satisfies our requirements (together with C).

Coanalytic hulls
In Theorems 4.1.1 and 4.1.8 we proved that the Borel hull in the definition of Haar null sets and Haar meager sets can be replaced by an analytic hull. The following theorems show that it cannot be replaced by a coanalytic hull. As the proofs of these theorems are relatively long, we do not reproduce them here. Note that these theorems were proved in the abelian case, but they can be generalized to the case when G is TSI.
In the case of Haar null sets, the paper [46] proves the following result: If G is a non-locally compact abelian Polish group, then there exists a coanalytic set A ⊆ G that is not Haar null, but there is a Borel probability measure μ on G such that μ(gAh) = 0 for every g, h ∈ G.
In particular, this set A is generalized Haar null, but not Haar null: This (and its generalization to TSI groups) is the best-known result for [46,Question 5.4], which asks the following: Theorem 4.2.4 (Doležal-Vlasák). If G is a non-locally compact abelian Polish group, then there exists a coanalytic set A ⊆ G that is not Haar meager, but there is a (nonempty) compact metric space K and a continuous function f :

Naive versions
It is possible to eliminate the Borel/universally measurable hull from our definitions completely. Unfortunately, the resulting 'naive' notions will not share the nice properties of a notion of smallness. The results in this subsection will show some of these problems. Most of these counterexamples are provided only in special groups or as a corollary of the Continuum Hypothesis, as even these 'weak' results are enough to show that these notions are not very useful.
The following two examples show that in certain groups the whole group is the union of countably many sets which are both naively Haar null and naively Haar meager, and hence neither the system of naively Haar null sets, nor the system of naively Haar meager sets is a σ-ideal. Example 4.3.3 [40]. Let G be an uncountable Polish group. Assuming the Continuum Hypothesis, there exists a subset W in the product group G × G such that both W and (G × G) \ W are naively Haar null and naively Haar meager.
Proof. Let < W be a well-ordering of G in order type ω 1 , and let W = {(g, h) ∈ G × G : g < W h} be this relation considered as a subset of G × G.
Let μ 1 be an nonatomic measure on G and μ 2 be a measure on G that is concentrated on a single point. It is clear that ( Let K ⊆ G be a nonempty perfect compact set (it is well known that such set exists) and Proof. The paper [34] constructs a decomposition of the plane with the following properties: Then the plane can be decomposed as R 2 =˙ i∈ω S i such that each line in the direction θ i intersects the set S i in at most one point.
We do not include the proof of this theorem. Later [47,Example 5.4] notices that the sets S i (i ∈ ω) in this decomposition are naively Haar null: Let μ i be the 1-dimensional Lebesgue measure on an arbitrary line with direction θ i . Then μ i (g + S i + h) = 0 for every g, h ∈ R 2 , because at most one point of g + S i + h is contained in the support of μ i . This means that S i is indeed naively Haar null.
A similar argument shows that these sets are also naively Haar meager: Let K i be a nonempty perfect compact subset of an arbitrary line with direction θ i , and let f i : K i → R 2 be the restriction of the identity function. Then f −1 i (g + S i + h) contains at most one point for every g, h ∈ R 2 , hence it is meager. This proves that S i is naively Haar meager.
The following theorem shows that in a relatively general class of groups the naive notions are strictly weaker than the corresponding 'canonical' notions. (1) There exists a subset of G that is naively Haar null but not Haar null.
(2) There exists a subset of G that is naively Haar meager but not Haar meager.
We do not reproduce the relatively long proof of these results. The proof of (1) can be found in [47,Theorem 1.3], the proof of (2) can be found in [38,Theorem 16]. Also note that when G is non-locally compact, these results are corollaries of Theorems 4.2.1 and 4.2.4.
The following example from [38,Proposition 17] yields the results of Example 4.3.3 in a different class of groups. The cited paper only proves this for the naively Haar meager case, but states that it can be proved analogously in the naively Haar null case. We do not reproduce this proof, as it is significantly longer than the proof of Example 4.3.3. Example 4.3.7 (Doležal-Rmoutil-Vejnar-Vlasák). Let G be an uncountable abelian Polish group. Assuming the Continuum Hypothesis, there exists a subset X ⊆ G such that both X and G \ X are naively Haar null and naively Haar meager.

Left and right Haar null sets
When we defined Haar null sets in Definition 3.1.1, we used multiplication from both sides by arbitrary elements of G. If we replace this by multiplication from one side, we get the following notions:  If 'Borel set' is replaced by 'universally measurable set', we can naturally obtain the generalized versions of these notions. As the papers about this topic happen to follow Christensen in defining 'Haar null set' to mean generalized Haar null set in the terminology of our paper, most results in this subsection were originally stated for these generalized versions.
There are situations where these 'one-sided' notions are more useful than the usual, symmetric one. (For example, [86] uses them to state results related to automatic continuity and the generalizations of the Steinhaus theorem. We mention some of these results in Subsection 5.2.) However, in some groups these are not 'good' notions of smallness in the sense of Subsection 3.2.
Translation invariance is not problematic even for the 'one-sided' notions: Suppose that, for example, B is Borel left Haar null and a Borel probability measure μ satisfies μ(gB) = 0 for every g ∈ G. If we consider a right translate Bh (this is the interesting case, invariance under left translation is trivial), then μ (X) = μ(Xh −1 ) is a Borel probability measure which satisfies μ (g · Bh) = 0 for every g ∈ G. An analogous argument shows that the system of right Haar null sets is translation invariant; using Proposition 4.4.8 it follows from these that the system of left-and-right Haar null sets is also translation invariant. It is clear that this reasoning works for the generalized versions, too.
Unfortunately there are Polish groups where these notions fail to form σ-ideals. In [86] Solecki gives a sufficient condition which guarantees that the generalized left Haar null sets form a σ-ideal and gives another sufficient condition which guarantees that the left Haar null sets do not form a σ-ideal. We state these results and show examples of groups satisfying these conditions without proofs: (Solecki). A Polish group G is called amenable at 1 if for any sequence (μ n ) n∈ω of Borel probability measures on G with 1 G ∈ supp μ n , there are Borel probability measures ν n and ν such that This class is closed under taking closed subgroups and continuous homomorphic images. A relatively short proof also shows that (II) is equivalent to This means that if G is amenable at 1, then so is the opposite group G opp . (G opp is the set G considered as a group with (x, y) → y · x as the multiplication.) Examples of groups which are amenable at 1 include: (1) abelian Polish groups; (2) locally compact Polish groups; (3) countable direct products of locally compact Polish groups such that all but finitely many factors are amenable; (4) inverse limits of sequences of amenable, locally compact Polish groups with continuous homomorphisms as bonding maps.
Theorem 4.4.4 (Solecki). If G is amenable at 1, then the generalized left Haar null sets form a σ-ideal.
Our note about the opposite group (and the fact that the intersection of two σ-ideals is also a σ-ideal) means that generalized right Haar null sets and generalized left-and-right Haar null sets also form σ-ideals.
The following definition is the sufficient condition for the 'bad' case. Note that this condition is also symmetric (in the sense that if G satisfies it, then G opp also satisfies it).
Definition 4.4.5 (Solecki). A Polish group G is said to have a free subgroup at 1 if it has a nondiscrete free subgroup whose all finitely generated subgroups are discrete.
The paper [86] lists several groups which all have a free subgroup at 1, we mention some of these: (1) countably infinite products of Polish groups containing discrete free nonabelian subgroups; (2) S ∞ , the group of permutations of N with the topology of pointwise convergence; (3) Aut(Q, ), the group of order-preserving self-bijections of the rationals with the topology of pointwise convergence on Q viewed as discrete (that is, a sequence (f n ) n∈ω ∈ G ω is said to be convergent if for every q ∈ Q there is a n 0 ∈ ω such that the sequence (f n (q)) n n0 is constant); Theorem 4.4.6 (Solecki). If G has a free subgroup at 1, then there are a Borel left Haar null set B ⊆ G and g ∈ G such that B ∪ Bg = G. As the left Haar null sets are translation invariant and G is not left Haar null, this means that they do not form an ideal.
As the notion of having a free subgroup at 1 is symmetric, the same is true for right Haar null sets.
In abelian groups it is trivial that the notions introduced by Definitions 4.4.1 and 4.4.2 are all equivalent to the notion of Haar null sets (and the generalized versions are equivalent to generalized Haar null sets).
The following remark is also trivial, but sometimes useful: , then gAh = gh · h −1 Ah = ghA (and similarly gAh = Agh) for every g, h ∈ G, hence if A is left (or right) Haar null, then A is Haar null.
For locally compact groups Theorem 3.3.7 and the remark in its proof show that all generalized right Haar null sets are Haar null. This implies that in locally compact groups the eight variants of Haar null (generalized or not, left or right or left-and-right or 'plain') are all equivalent to having Haar measure zero.
The following diagram summarizes the situation in the general case: (Here means that there is a counterexample in a suitable Polish group and ⇒ means that the implication is true in all Polish groups.) The not-implications marked by (1) are demonstrated by the already stated Theorem 4.4.6. We will prove the implication (2) Define then μ is clearly a Borel probability measure and if the characteristic function of a set S is denoted by χ S , then using Fubini's theorem we have and these show that μ satisfies the requirements of Definition 4.4.2.
The following two counterexamples appear in [87]. This paper uses relatively elementary techniques and examines the structure of the group (This is the group of order-preserving self-homeomorphisms of [0,1], the group operation is composition, the topology is the compact-open topology.) We state these results without proofs:    (Solecki). Let G be an amenable at 1 Polish group. Do left Haar null subsets of G coincide with right Haar null subsets of G?
We conclude this part by stating [85, Theorem 6.1], which provides a necessary and sufficient condition for the equivalence of the notions generalized left Haar null and generalized Haar null in a special class of groups.
Theorem 4.4.12 (Solecki). Let H n (n ∈ ω) be countable groups and consider the group G = n H n . The following conditions are equivalent.
(1) In G the system of generalized left Haar null sets is the same as the system of generalized Haar null sets.
(2) For each universally measurable set A ⊆ G that is not generalized Haar null, (4) For all but finitely many n ∈ ω all elements of H n have finite conjugacy classes in H n , that is, for all but finitely many n ∈ ω and for all x ∈ H n the set {yxy −1 : y ∈ H n } is finite.

Openly Haar null sets
Definition 4.5.1. A set A ⊆ G is said to be openly Haar null if there is a Borel probability measure μ on G such that for every ε > 0 there is an open set U ⊇ A such that μ(gU h) < ε for every g, h ∈ G. If μ has these properties, we say that μ witnesses that A is openly Haar null.
The notion of openly Haar null sets was introduced in [84], and more thoroughly examined in [26].
The following simple proposition shows that this is a stronger property than Haar nullness: Proof. For every n ∈ ω there is an open set U n ⊇ A such that μ(gU n h) < 1 n+1 for every g, h ∈ G. Then B = n∈ω U n is a G δ set that is Haar null because it satisfies μ(gBh) μ(gU n h) < 1 n+1 for every g, h ∈ G, hence μ(gBh) = 0 for every g, h ∈ G.
Remark 4.5.3. In non-locally compact abelian Polish groups 'openly Haar null' is a strictly stronger property than 'Haar null', because [46,Theorem 1.3] states that in these groups there is a Haar null set that is not contained in a G δ Haar null set.
To prove that the system of openly Haar null sets forms a σ-ideal, we will need the following technical lemma: Lemma 4.5.4. If μ witnesses that A ⊆ G is openly Haar null and V is a neighborhood of 1 G , then there exists a measure μ which also witnesses that A is openly Haar null and has a compact support that is contained in V .
Proof. Lemma 3.2.2 states that there are a compact set C ⊆ G and c ∈ G such that μ(C) > 0 and C ⊆ cV . Let μ (X) := μ(cX∩C) μ(C) , this is clearly a Borel probability measure and has a compact support that is contained in V . Fix an arbitrary ε > 0. We will find an open set U ⊇ A such that μ (gU h) < ε for every g, h ∈ G. Note that There exists an open set U ⊇ A with μ(gU h) < μ(C) · ε for every g, h ∈ G, and this satisfies μ(cgU h) < μ(C) · ε for every g, h ∈ G, hence μ has the required properties. Proof. It is trivial that the system of openly Haar null sets satisfies (I) and (II) in Definition 3.2.1. The following proof of (III) is described in the Appendix of [26] and is similar to the proof of Theorem 3.2.5.
Let A n be openly Haar null for all n ∈ ω, we prove that A = n∈ω A n is also openly Haar null. For every set A n fix a measure μ n which witnesses that A n is openly Haar null. Let d be a complete metric on G that is compatible with the topology of G.
We construct for all n ∈ ω a compact set C n ⊆ G and a Borel probability measureμ n such that the support ofμ n is C n ,μ n (gU n h) < ε for every g, h ∈ G and the 'size' of the sets C n decreases 'quickly'.
The construction will be recursive. For the initial step use Lemma 4.5.4 to find a measurẽ μ 0 witnessing that A 0 is openly Haar null and has compact support C 0 ⊆ G. Assume thatμ n and C n are already defined for all n < n. By Lemma 3.2.4 there exists a neighborhood V n of 1 G such that if v ∈ V n , then d(k · v, k) < 2 −n for every k in the compact set C 0 C 1 C 2 · · · C n−1 . Applying Lemma 4.5.4 again we can find a measureμ n with compact support C n ⊆ V n which is witnessing that A n is openly Haar null.
If c n ∈ C n for all n ∈ ω, then it is clear that the sequence (c 0 c 1 c 2 · · · c n ) n∈ω is a Cauchy sequence. As (G, d) is complete, this Cauchy sequence is convergent; we write its limit as the infinite product c 0 c 1 c 2 · · · . The map ϕ : n∈ω C n → G, ϕ((c 0 , c 1 , c 2 , . . .)) = c 0 c 1 c 2 · · · is the pointwise limit of continuous functions, hence it is Borel.
Let μ Π be the product of the measuresμ n on the product space C Π := n∈ω C n . Let μ = ϕ * (μ Π ) be the push-forward of μ Π along ϕ onto G, that is, We claim that μ witnesses that A = n∈ω A n is openly Haar null. Fix an arbitrary ε > 0, we will show that there is an open set U ⊇ A such that μ(gU h) < ε for every g, h ∈ G. It is enough to find open sets U n ⊇ A n such that μ(gU n h) < ε · 2 −(n+2) for every g, h ∈ G and n ∈ ω, because then U = n∈ω U n satisfies that for every g, h ∈ G Fix g, h ∈ G and n ∈ ω. Choose an U n ⊇ A n open set satisfying thatμ n (gU n h) < ε · 2 −(n+2) for every g, h ∈ G. Note that if c j ∈ C j for every j = n, j ∈ ω, theñ μ n ({c n ∈ C n : c 0 c 1 c 2 · · · c n · · · ∈ gU n h}) Applying Fubini's theorem in the product space ( j =n C j ) × C n to the product measure ( j =nμ j ) ×μ n yields that ε · 2 −(n+2) > μ Π (c 0 , c 1 , . . . , c n , . . .) ∈ C Π : c 0 c 1 · · · c n · · · ∈ gU n h .
Unfortunately, there are groups where this ideal contains only the empty set. The following results about this 'collapse' are proved as [26, Propositons 5 and 3]:  (1) G is uncountable and admits a two-sided invariant metric.
(2) G = S ∞ is the group of permutations of N with the topology of pointwise convergence.
(3) G = Aut(Q, ) is the group of order-preserving self-bijections of the rationals with the topology of pointwise convergence on Q viewed as discrete (that is, a sequence (f n ) n∈ω ∈ G ω is said to be convergent if for every q ∈ Q there is a n 0 ∈ ω such that the sequence (f n (q)) n n0 is constant).
(4) G = U ( 2 ) is the unitary group on the separable infinite-dimensional complex Hilbert space with the strong operator topology.
(5) G admits a continuous surjective homomorphism onto a group which has a nonempty openly Haar null subset.
In Subsection 5.4 we will mention Theorem 5.4.3, which is an interesting application of this notion. Dodos introduced this notion in the paper [35] (under the name 'strongly Haar null sets' and only considering abelian groups), and later generalized them for arbitrary Polish groups in [37] (where they are called 'generically Haar null sets').

Generically Haar null sets
The 'trick' of this definition is that the system of generically Haar null sets is obviously closed under countable unions because T ( n∈ω A n ) ⊇ n∈ω T (A n ) and the countable intersection of comeager sets is still comeager. The following proposition states this fact and the other useful properties which are evident from the definition: Proposition 4.6.2. The system of generically Haar null sets is translation-invariant and forms a σ-ideal in the σ-algebra of universally measurable sets. Every generically Haar null set is a generalized Haar null set.
Note that we only defined generically Haar null sets among the universally measurable sets. In the special case when G is abelian, this notion coincides with the 'generic' version of   It is also possible to define an analogous one-sided notion:  The following trichotomy gives additional motivation for this notion: Theorem 4.6.6 (Dodos). Each analytic set A ⊆ G satisfies one of the following.
(2) T (A) is meager and dense in P (G).
(3) T (A) is comeager in P (G) (that is, A is generically Haar null).
Proof. First we prove that if A is generalized Haar null, then T (A) is dense in P (G) (this part of the proof is from [36, Proposition 5 (i)]). Let δ x denote the Dirac measure concentrated at x ∈ G (that is, δ x (X) = 1 if x ∈ X and δ x (X) = 0 otherwise). The system is a dense subset of P (G) (see [67,Theorem 17.19]). It is enough to prove that if ϕ = n i=1 α i δ xi ∈ F is arbitrary, then any of its neighborhoods contains an element of T (A).
Theorem 4.6.8 (Dodos). Let X be a Polish space and F be a face of P (X) with the Baire property. Then F is either a meager or a comeager subset of P (X).
The proof of this result can be found as [35,Theorem B]. It is easy to see that if A is a universally measurable generalized Haar null set, then T (A) is a face of P (G).
To finish the proof of the theorem it is enough to prove the following claim: Proof. It is clear that the set is analytic. If we apply Theorem 4.1.3 for the set S, X = G × G and Y = G, then we can see that is analytic. If we take the section of this set at r = 0 and project this section on oP (G), then we can see that is analytic, T (A) is coanalytic. It follows from [67, Corollary 29.14] that analytic and coanalytic sets have the Baire property, concluding the proof.
It is clear that the following one-sided variant of this trichotomy can be proved analogously: Theorem 4.6.10 (Dodos). Each analytic set A ⊆ G satisfies one of the following.
(1) T l (A) = ∅ (that is, A is not generalized left Haar null).
(2) T l (A) is meager and dense in P (G).
(3) T l (A) is comeager in P (G) (that is, A is generically left Haar null).
Following [35,36], we stated the trichotomy Theorem 4.6.6 for analytic sets. This is formally stronger than stating it for Borel sets, but it turns out that the analytic generically Haar null sets are just the analytic subsets of the Borel generically Haar null sets: Theorem 4.6.11 (Dodos). Let A ⊆ G be an analytic generalized Haar null set. Then there exists a Borel Haar null Proof. This result originally appears as [36,Corollary 12], in a paper which only considers the question in the abelian case. We use a different method to prove this result (in all Polish groups G); this method is from [83] where it was used to prove the result which we stated as the equivalence (1) ⇔ (5) in Theorem 4.1.1.
In the proof of Theorem 4.6.6 we proved Claim 4.6.9 which states that T (A) has the Baire property. This implies that T (A) can be written as T (A) = R ∪ M where R is a nonempty Borel subset of P (G) and M is meager in P (G). is coanalytic on analytic, that is, for every Polish space Y and P ∈ Σ 1 1 (Y × G), the set {y ∈ Y : P y ∈ Φ} is Π 1 1 .
Proof. Let Y be Polish space and P ∈ Σ 1 1 (Y × G) and let P = {(g, h, y, γ ThenP is analytic, as it is the preimage of P under (g, h, y, γ) → (y, g −1 γh −1 ). Applying Theorem 4.1.3 yields that {(μ, g, h, y, r) ∈ P (G) × G × G × Y × R : μ(P (g,h,y) ) > r} is analytic, therefore its section is also analytic. If we intersect this with the Borel set R × G × G × Y , we get the analytic set Projecting this on Y yields that {y : μ(P (g,h,y) ) > 0 for some g, h ∈ G and μ ∈ R} is analytic. Using this and the fact thatP (g,h,y) = gP h y (by definition) yields that {y ∈ Y : μ(P (g,h,y) ) = 0 for all g, h ∈ G and μ ∈ R} = {y ∈ Y : P y ∈ Φ} is coanalytic. The following result is an immediately corollary of this theorem:

An analytic set A ⊆ G is generically Haar null if and only if there is a Borel generically Haar null set B ⊆ G such that A ⊆ B.
It is relatively easy to prove that a Borel generically Haar null set is not only Haar null, but also meager. (This idea appears as part (ii) of [36,Proposition 5].) In fact, this is also true for generically left Haar null sets: Proposition 4.6.14 (Dodos). If B is a Borel generically left Haar null set, then B is meager.
Proof. Assume for the contrary that B is not meager. This implies that B is comeager in some open subset of G (see, for example, [67, 8.26]). If D is a countable dense subset of G, then the product DB = {db : d ∈ D, b ∈ B} is comeager in G. It follows from [21, Theorem 10] that μ(DB) = 1 for the generic measure μ ∈ P (G). On the other hand, the generic measure μ ∈ P (G) satisfies that μ(gB) = 0 for all g ∈ G (because B is generically left Haar null), thus μ(DB) g∈D μ(gB) = 0, a contradiction. Note that this proof works even if we replace 'Borel' with 'universally measurable and with the Baire property' in the proposition.
The following proposition implies that this σ-ideal is nontrivial (contains nonempty sets) in abelian groups: Proposition 4.6.15 (Dodos)

. Assume that G is an abelian Polish group. If A ⊂ G is a σ-compact Haar null set, then A is generically Haar null.
The proof of this result can be found as part (iii) of [36,Proposition 5]. Note that this proposition implies that if (the abelian Polish group) G is locally compact, then the closed Haar null sets are generically Haar null. The converse of this is also true: Theorem 4.6.16 (Dodos)

. Assume that G is an abelian Polish group. Then G is locally compact if and only if every closed Haar null set is generically Haar null.
The proof of this result can be found as [35,Corollary 9]. In Subsection 5.2 we will state a variant of the Steinhaus theorem (Theorem 5.2.9) which uses the generically left Haar null sets as small sets.
For more information about related notions in the special case of abelian Polish groups, see [9].

Strongly Haar meager sets
Strongly Haar meager sets are the variant of Haar meager sets where we require the witness function to be the identity function of G restricted to a compact subset C ⊆ G. (Despite the similar name, this notion is unrelated to the generically Haar null sets, which are also called strongly Haar null sets in [35].) This is motivated by the fact that when we prove that some set is Haar meager, we frequently use witness functions of this kind.
Definition 4.7.1. A set A ⊆ G is said to be strongly Haar meager if there are a Borel set B ⊇ A and a (nonempty) compact set C ⊆ G such that gBh ∩ C is meager in C for every g, h ∈ G.
The following basic question is [29, Problem 2]: Question 4.7.2 (Darji). Is every Haar meager set strongly Haar meager? (Note that the paper [29] only considered the case of abelian groups.) The result [9,Theorem 5.13] shows that in a certain class of abelian Polish groups the Haar meager sets and the strongly Haar meager sets coincide:   Theorem 4.7.6 (Elekes-Nagy-Poór-Vidnyánszky). In the abelian Polish group Z ω , there exists a G δ set that is Haar meager but not strongly Haar meager.

The other basic question of Darji related to strongly Haar meager sets is [29, Problem 3]:
Question 4.7.7 (Darji). Is the system of strongly Haar meager sets a σ-ideal?
(Note that while the paper [29] only considered the case of abelian groups, this question seems to be interesting in other Polish groups as well.) Even the following variant of this question seems to be open:  (2) In the Polish space K(G) of nonempty, compact subsets of G

{K ∈ K(G) : K witnesses that B is strongly Haar meager} is comeager.
Sets that satisfy these equivalent conditions are called generically Haar meager sets; it is easy to see that the system of these sets is closed under countable unions.

Aronszajn, cube, and Gauss null sets
In separable Banach spaces, there are various other notions of nullsets. Let us describe some of these very briefly.
A Borel measure μ on a separable (real) Banach space X is called a cube measure if it is the push-forward of the standard measure (the product of the Lebesgue measures) on [0, 1] N along the map r → x + i r i x i , where x, x i ∈ X are fixed and i |x i | < ∞. A Borel measure μ on X is called Gaussian if its push-forward μ • (x * ) −1 is a Gaussian measure (the distribution of a normal random variable) on R for every x * ∈ X * . A Borel measure is called nondegenerate if all closed hyperplanes are of measure zero. The fundamental result in this area is Csörnyei's theorem stating that these three notions actually coincide [27]. As for the connection with Haar null sets, it is clear from the definition that all cube null sets are Haar null. However, the converse implication does not hold in general, as Theorem 6.4.2 implies that the compact sets are Haar null in a separable Banach space, but one can show that there are compact sets that are not cube null. For more information on these notions, especially for the connection between these notions and differentiability of Lipschitz functions, see, for example, [79].
It is also worth mentioning that there is a further important related notion, that of the so-called Γ-null sets, but we do not give the somewhat technical definition here. This can also be found in [79].

Analogs of the results from the locally compact case
In this section we discuss generalizations and analogs of a few theorems that are well known for locally compact groups. Unfortunately, although these are true in locally compact groups for the sets of Haar measure zero and the meager sets, neither of them remains completely valid for Haar null sets and Haar meager sets. However, in some cases weakened versions remain true, and these often prove to be useful.

Fubini's theorem and the Kuratowski-Ulam theorem
Fubini's theorem, and its topological analog, the Kuratowski-Ulam theorem (see, for example, [67, 8.41]) describes small sets in product spaces. They basically state that a set (which is measurable in the appropriate sense) in the product of two spaces is small if and only if cosmall many sections of it are small. Note that because the product of left (or right) Haar measures of two locally compact groups of is trivially a left (or right) Haar measure of the product group, (a special case of) Fubini's theorem connects the sets of Haar measure zero in the two groups and the sets of Haar measure zero in the product group.
Unfortunately, analogs of these theorems are proved only in very special cases, and there are counterexamples known in otherwise 'nice' groups. We provide a simple counterexample (which can be found as [38,Example 20]) that works in both the Haar null and Haar meager case.
Example 5.1.1 (folklore). There exists a closed set A ⊆ Z ω × Z ω that is neither Haar null nor Haar meager, but in one direction all its sections are Haar null and Haar meager. (In the other direction, non-Haar-null and non-Haar-meager many sections are non-Haar-null and non-Haar-meager.) Proof. The group operation of Z ω × Z ω is denoted by +. The set with these properties will be A = {(s, t) ∈ Z ω × Z ω : t n s n 0 for every n ∈ ω}.
It is clear from the definition that A is closed.
Note that for t ∈ Z ω , the section A t = {s ∈ Z ω : t n s n 0 for every n ∈ ω} is compact (as it is the product of finite sets with the discrete topology), and it follows from Theorem 6.4.2 and Corollary 6.4.7 that all compact sets are Haar null and Haar meager in Z ω .
To show that A is not Haar null and not Haar meager, we will use the technique described in Subsection 6.6 and show that for every compact set C ⊆ Z ω × Z ω the set A contains a translate of C. As C = ∅ satisfies this, we may assume that C = ∅. Let π 1 n (s, t) = s n and π 2 n (s, t) = t n , then π 1 n , π 2 n : Z ω × Z ω → Z are continuous functions. Let a and b be the sequences satisfying a n = − min π 1 n (C) and b n = − min π 2 n (C) + max π 1 n (C) − min π 1 n (C). It is straightforward to check that this choice guarantees that if (s, t) ∈ C + (a, b), then t n s n 0 for every n ∈ ω.
Finally, if s ∈ Z ω satisfies s n 0 for every n ∈ ω, then similar, but simpler arguments show that the section A s = {t ∈ Z ω : t n s n for every n ∈ ω} contains a translate of every compact set C ⊆ Z ω , hence it is neither Haar null nor Haar meager. Similarly, the set {s ∈ Z ω : s n 0 for every n ∈ ω} is also neither Haar null nor Haar meager, so we proved the statement about the sections in the other direction.
The following counterexample appears as [23, Theorem 6]: Example 5.1.2 (Christensen). Let H be a separable infinite-dimensional Hilbert space (with addition as the group operation) and let S 1 be the unit circle in the complex plane (with complex multiplication as the group operation). There exists in the product group H × S 1 a Borel set A such that (I) for every h ∈ H, the section A h has Haar measure one in S 1 ; (II) for every s ∈ S 1 , the section A s is Haar null in H; (III) the complement of A is Haar null in the product group H × S 1 .
Note that here S 1 is a compact group. The connection between (I) and (III) in this example is not accidental: ·) is an abelian Polish group and (G 2 , ·) is a locally compact abelian Polish group and A ⊂ G 1 × G 2 is universally measurable, then the following are equivalent.
(2) There is a generalized Haar null set N ⊂ G 1 such that for all g 1 ∈ G 1 \ N , the section A g1 is Haar null in G 2 (that is, the Haar measure on G 2 assigns measure zero to it).
This connection is mentioned in [23] and proved in [18, Theorem 2.3]. We prove the following generalized version of this result: (1) There exists a Haar null set E ⊆ H and a Borel probability measure μ on G such that every h ∈ H \ E and g 1 , g 2 ∈ G satisfies μ(g 1 B h g 2 ) = 0 (that is, these sections are Haar null and μ is witness measure for each of them).

(2) B is Haar null in G × H. (3) There exists a Haar null set E ⊆ H such that for every h ∈ H \ E the section B h has Haar measure zero in G.
Proof. (1) ⇒ (2): As the set E is Haar null, there exists a Borel set E ⊇ E and a Borel probability measure ν on H such that ν(h 1 E h 2 ) = 0 for every h 1 , h 2 ∈ H. We show that the measure μ × ν witnesses that B is Haar null in G × H. Fix arbitrary g 1 , g 2 ∈ G and h 1 , h 2 ∈ H, we have to prove that (μ × ν)((g 1 , h 1 ) · B · (g 2 , h 2 )) = 0.
Note that and in this union the (μ × ν)-measure of the first term is zero (by the choice of μ and E ). For every h ∈ H, the h-section of the Borel set ( · g 2 , and this is either the empty set (if h −1 . Applying Fubini's theorem in the product space G × H to the product measure μ × ν yields that (μ × ν)((g 1 , h 1 ) · B · (g 2 , h 2 )) = 0, and this is what we had to prove.
(2) ⇒ (3) (when G is locally compact): Let B ⊆ G × H be Haar null and suppose that μ is a witness measure, that is, μ((g 1 , h 1 ) · B · (g 2 , h 2 )) = 0 for every g 1 , g 2 ∈ G and h 1 , h 2 ∈ H. Fix a left Haar measure λ on G, let δ denote the Dirac measure at 1 H (that is, for X ⊆ H, δ(X) = 1 if 1 H ∈ X and 0 otherwise) and letλ = λ × δ. Let E = {h ∈ H : λ(B h ) = 0}, it is clearly enough to prove that E ⊆ H is Haar null. First we use a standard argument to show that E is a Borel set. is a Borel probability measure on G that is equivalent to λ in the sense that they have the same zero sets, then E =Ẽ is Borel, because it is the section of a Borel set.
Consider the set it is easy to see that this is a Borel set. Applying Fubini's theorem in the product space because μ witnesses that B is Haar null. Applying Fubini's theorem again for the other direction yields that It is clear from the definition of E that the function takes strictly positive values on the set G × (h 1 Eh 2 ). However, our calculations showed that ϕ d μ is zero, hence the μ-measure of the set G × (h 1 Eh 2 ) must be zero. (3) states that λ(B h ) = 0 for every h ∈ H \ E. Using Theorem 3.3.5 it is easy to see that λ(g 1 B h g 2 ) = 0 for every g 1 , g 2 ∈ G. Let C ⊆ G be a Borel set such that 0 < λ(C) < ∞ (the regularity of the Haar measure guarantees a compact C satisfying this) and let μ be the Borel probability measure μ(X) = λ(X∩C) λ(C) . Then μ λ guarantees that μ satisfies condition (1).
We also prove the analog of this for Haar meager sets: (1) There exists a Haar meager set E ⊆ H, a (nonempty) compact metric space K and a continuous function f : K → G such that every h ∈ H \ E and g 1 , g 2 ∈ G satisfies that f −1 (g 1 B h g 2 ) is meager in K (that is, these sections are Haar meager and f is a witness function for each of them).
(2) B is Haar meager in G × H.

(3) There exists a Haar meager set E ⊆ H such that for every h ∈ H \ E the section B h is meager in the locally compact group G.
Proof. (1) ⇒ (2): As the set E is Haar meager, there exist a Borel set E ⊇ E, a (nonempty) compact metric space K and a continuous function ϕ : . We show that the function f × ϕ witnesses that B is Haar meager in G × H. Fix arbitrary g 1 , g 2 ∈ G and Note that and in this union and using the choice of ϕ and the Kuratowski-Ulam theorem it is clear that this is a meager subset of K × K . For every h ∈ H, the h-section of the Borel set ( · g 2 , and this is either the empty set (if h −1 1 hh −1 2 ∈ E ) or a set whose preimage under f is meager in . This means that for every k ∈ K , the k -section of the set Applying the Kuratowski-Ulam theorem in the product space K × K yields that the set (f × ϕ) −1 ((g 1 , h 1 ) · B · (g 2 , h 2 )) is indeed meager in K × K , and this is what we had to prove.
(2) ⇒ (3) (when G is locally compact): Let B ⊆ G × H be Haar meager and suppose that f : K → G × H is a witness function (where K is a nonempty compact metric space), that is, f −1 ((g 1 , h 1 ) · B · (g 2 , h 2 )) is meager in K for every g 1 , g 2 ∈ G and h 1 , and the function ψ : G → G × H, ψ(g) = (g, 1 H ) (ψ is the analog ofλ from the case of measure).
and this is meager in K (as f witnesses that B is Haar meager), hence the Kuratowski-Ulam theorem yields that (f × ψ) −1 (S) is meager in K × G.
Applying the Kuratowski-Ulam theorem in the other direction yields that the set Thus we know that is not meager in G} is meager in K, and because meagerness is translation invariant is not meager in G} is meager in K.
But note that is not meager in G} so we proved that f H is indeed a witness measure for E.
(3) ⇒ (1) (when G is locally compact): Note that the proof of Theorem 3.3.13 shows that in a locally compact Polish group every meager set is Haar meager and there is a function which is a witness function for each of them. The implication that we have to prove is clearly a special case of this observation.
Finally, we state another special case when the analog of the Kuratowski-Ulam theorem is valid. The proof of this result can be found in [38,Theorem 18].

The Steinhaus theorem
The Steinhaus theorem and its generalizations state that if a set A ⊆ G is not small (and satisfies some measurability condition, for example, it is in the σ-ideal generated by the Borel sets and the small sets), then AA −1 = {xy −1 : x, y ∈ A} contains a neighborhood of 1 G .
The original result of Steinhaus stated this in the group (R, +) and used the sets of Lebesgue measure zero as the small sets. Weil extended this for an arbitrary locally compact group, using the sets of Haar measure zero as the small sets. We already proved this result and used it to prove that there are no (left or right) Haar measures in non-locally compact groups.
Remark 5.2.1. The proof of this result also works for non-locally compact groups, but in those groups the result is vacuously true.
It is natural to ask the following question (a question of this type appears as, for example, [ One of these is stated as [81, Theorem 2.8] and is a slight variation of a result in [22]. This result does not claim that AA −1 will be a neighborhood of 1 G , only that finitely many conjugates of it will cover a neighborhood of 1 G . Also, it uses generalized right Haar null sets as small sets (see Subsection 4.4 for the definition), which is a weaker notion than generalized Haar null sets (hence this result does not imply the variant where 'right' is omitted from the text).
Theorem 5.2.3 (Christensen, Rosendal). Suppose that A ⊆ G is a universally measurable subset which is not generalized right Haar null. Then for any neighborhood W of 1 G there are n ∈ ω and h 0 , h 1 , h 2 Proof. Suppose that the conclusion fails for A and W , that is, for every n ∈ ω and h 0 , h 1 , . . . h n−1 ∈ W and any neighborhood V 1 G , there is some Then we can inductively choose a sequence (g j ) j∈ω such that g j → 1 G and for every j 0 < j 1 < j 2 < . . . index sequence and r ∈ ω (I) the infinite product g j0 g j1 g j2 · · · converges (this can be achieved, for example, by requiring d(g j0 · · · g jr−1 , g j0 · · · g jr−1 · g jr ) < 2 −jr where d is a fixed complete metric on G); (II) g jr / ∈ (g j0 · · · g jr−1 ) −1 AA −1 (g j0 · · · g jr−1 ).
Using (I) we can define a continuous map ϕ : 2 ω → G by where g 0 = 1 G and g 1 = g. Let λ be the Haar measure on the Cantor group (Z 2 ) ω = 2 ω and note that as A is not generalized right Haar null, there is some g ∈ G such that λ(ϕ −1 (Ag)) = ϕ * (λ)(Ag) > 0 So by Corollary 3.3.10, contains a neighborhood of the identity (0, 0, . . .) in 2 ω . This neighborhood must contain an element with exactly one '1' coordinate, so there are α, β ∈ ϕ −1 (Ag) and m ∈ ω such that α(m) = 1, β(m) = 0, and α(j) = β(j) for every j ∈ ω, j = m. This means that there are h = g j0 g j1 · · · g jr−1 , j 0 < j 1 < · · · < j r−1 < m and k ∈ G such that ϕ(α) = hg m k and ϕ(β) = hk. It follows that The following special case is often useful: Corollary 5.2.4. Suppose that A ⊆ G is a universally measurable subset which is conjugacy invariant (that is, gAg −1 = A for every g ∈ G; in abelian groups every subset has this property) and not generalized Haar null. Then AA −1 is a neighborhood of the identity.
Proof. If A is conjugacy invariant, then as we noted, gAh = gAg −1 · gh = Agh and thus A is generalized right Haar null if and only if A is generalized Haar null. Moreover, if A is conjugacy invariant, then gAA −1 g −1 = (gAg −1 ) · (gAg −1 ) −1 = AA −1 , hence AA −1 is also conjugacy invariant and thus in this case Theorem 5.2.3 states that AA −1 is a neighborhood of identity.
Variants of the Steinhaus theorem can be used to prove results about automatic continuity (results stating that all homomorphisms π : G → H which satisfy certain properties are continuous). For example, the classical result Corollary 3.3.10 yields that any universally measurable homomorphism from a locally compact Polish group into another Polish group is continuous (see, for example, [81,Corollary 2.4]; a map is said to be universally measurable if the preimages of open sets are universally measurable). We prove the following automatic continuity result as a corollary of Theorem 5.2.3.  (1 H , r) are conjugacy invariant sets and form a neighborhood base of 1 H , hence we may also assume that U is conjugacy invariant.
G can be covered by countably many right translates of π −1 (U ), because there is a countable set S ⊆ π(G) that is dense in π(G) and thus the system {U · s : s ∈ S} covers π(G). This means that the universally measurable set π −1 (U ) is not right Haar null, hence we may apply Theorem 5.2.3 to see that for some n ∈ ω and h 0 , h 1 , . . . , h n−1 ∈ G the set is a neighborhood of 1 G . For every g ∈ W there is a 0 j < n such that g ∈ h j π −1 (U )(π −1 (U )) −1 h −1 j , but then thus π(W ) ⊆ UU −1 ⊆ V and this shows that π is continuous at 1 G .
In [86] Solecki proved that if G is amenable at 1 (see Definition 4.4.3), then a simpler variant of the Steinhaus theorem is true, but if G has a free subgroup at 1 (see Definition 4.4.5) and satisfies some technical condition, then this variant is false. The following theorems state these results.
Theorem 5.2.6 (Solecki). If G is amenable at 1 and A ⊆ G is universally measurable and not generalized left Haar null, then A −1 A contains a neighborhood of 1 G .
Definition 5.2.7. A Polish group G is called strongly non-locally compact if for any neighborhood U of 1 G there exists a neighborhood V of 1 G such that U cannot be covered by finitely many sets of the form gV h with g, h ∈ G.
Note that the examples we mentioned after Definition 4.4.5 are all strongly non-locally compact.

Theorem 5.2.8 (Solecki). If G has a free subgroup at 1 and is strongly non-locally compact, then there is a Borel set A ⊆ G which is not left Haar null and satisfies 1 G / ∈ int(A −1 A).
Using the notion of generically left Haar null sets (which we describe in Subsection 4.6), it is possible to prove another weakened version of Question 5.2.2: Theorem 5.2.9 (Dodos). Suppose that A ⊆ G is an analytic set which is not generically left Haar null. Then A −1 A is not meager in G.
The relatively long proof of this result can be found as [37,Theorem A]. Combining this and Pettis' theorem yields the following corollary: Corollary 5.2.10 (Dodos). If A ⊆ G is analytic and not generically left Haar null, then The above mentioned Pettis' theorem (which is also known as Piccard's theorem and appears as, for example, [68, Theorem 2.9.6] and [67, Theorem 9.9]) is the analog of the Steinhaus theorem which uses the meager sets as small sets.
As 'Haar meager' is a stonger notion than 'meager', the following result strengthens this classical result in the abelian case: Theorem 5.2.11 (Jab lońska). Let G be an abelian Polish group. If A ⊆ G is a Borel set that is not Haar meager, then AA −1 is a neighborhood of 1 G .
The proof of this theorem can be found in [63]; the method of the proof is similar to that of Theorem 5.2.3.
For more information about this question in the special case of abelian Polish groups, see the paper [9].

The countable chain condition
The countable chain condition (often abbreviated as ccc) is a well-known property of partially ordered sets or structures with associated partially ordered sets. A partially ordered set (P, ) is said to satisfy the countable chain condition if every strong antichain in P is countable. (A set A ⊆ P is a strong antichain if ∀x, y ∈ A : (x = y ⇒ z ∈ P : (z x and z y)).) This is called countable 'chain' condition because in some particular cases this condition happens to be equivalent to a condition about lengths of certain chains.
We will apply the countable chain condition to notions of smallness in the following sense: Suppose that X is a set and S ⊆ A ⊆ P(X). We say that S has the countable chain condition in A if there is no uncountable system U ⊆ A \ S such that U ∩ V ∈ S for any two distinct U, V ∈ U.
In our cases A will be a σ-algebra and S will be the σ-ideal of 'small' sets. Note that S has the countable chain condition in A if and only if the partially ordered set (A \ S, ⊆) satisfies the countable chain condition. Also note that ifS ⊆ S andÃ ⊇ A, then 'S has the countable chain condition inÃ' is a stronger statement than 'S has the countable chain condition in A'.
We will generalize the following two classical results (these are stated as [67, Exercises 17.2 and 8.31]). Proposition 5.3.2. If μ is a σ-finite measure, the σ-ideal of sets with μ-measure zero has the countable chain condition in the σ-algebra of μ-measurable sets.
Proposition 5.3.3. In a second countable Baire space, the σ-ideal of meager sets has the countable chain condition in the σ-algebra of sets with the Baire property.
The theorems in Subsection 3.3 state that if G is locally compact, then N = HN = GHN (that is, of Haar measure zero ⇔ Haar null ⇔ generalized Haar null) and M = HM (that is, meager ⇔ Haar meager). The special case of Proposition 5.3.2 where μ = λ for a left Haar measure λ on G means that the σ-ideal HN = GN H has the countable chain condition in the σ-algebra of λ-measurable sets. As every universally measurable set is λ-measurable, this clearly implies that HN = GHN has the countable chain condition in the σ-algebra of universally measurable sets. Analogously, Proposition 5.3.3 means that in a locally compact Polish group G the σ-ideal HM has the countable chain condition in the σ-algebra of sets with the Baire property.
For the case of measure Christensen asked in [23,Problem 2] whether it this true in the non-locally compact case (the paper [23] considers only abelian groups, but the problem is interesting in general).
The following simple example shows that the answer for this is negative in the group Z ω (a variant of this is stated in [40,Proposition 1]). It also answers the analogous question in the case of category. Proof. It is clear that S(A) is closed for every A ⊆ ω. If A and B are two different subsets of ω, then some n ∈ ω satisfies, for example, n ∈ A \ B and thus ∀s ∈ S(A) : s n 0, but ∀s ∈ S(B) : s n < 0.
Finally, for every A ⊆ ω the set S(A) contains a translate of every compact subset C ⊆ Z ω , because if we define t (C) ∈ Z ω by then clearly C + t (C) ⊆ A. Applying Lemma 6.6.2 concludes our proof.
In [83] Solecki showed that the situation is the same in all non-locally compact groups that admit a two-sided invariant metric. This is a corollary of Theorem 3.3.15, which we already stated without proof. As we will use Lemma 6.6.2 again, this also answers the question in the case of category. Example 5.3.5 (Solecki). Suppose that G is non-locally compact and admits a two-sided invariant metric. Then none of HN , GHN , and HM has the countable chain condition in B(G).
Proof. By Theorem 3.3.15 there exists a closed set F ⊆ G and a continuous function ϕ : F → 2 ω such that for any x ∈ 2 ω and any compact set C ⊆ G there is a g ∈ G with gC ⊆ ϕ −1 ({x}). Then the system {ϕ −1 ({x}) : x ∈ 2 ω } consists of continuum many pairwise disjoint closed sets and they all satisfy the requirements of Lemma 6.6.2, hence they are neither generalized Haar null nor Haar meager.

Decomposition into a Haar null and a meager set
In a locally compact group G the regularity of the Haar measures implies that the group can be written as G = N ∪ M where N is of Haar measure zero and M is meager.
In [28] Darji asks if this holds for non-locally compact groups: Question 5.4.1 (Darji). Can every uncountable Polish group be written as the union of two sets, one meager and the other Haar null?
Remark 5.4.2. More precisely, [28] asks whether every uncountable Polish group can be written as the union of a meager and a generalized Haar null set (in the terminology of this paper). However, it is easy to see that this distinction is inconsequential, as every meager set is contained in an F σ (in particular, Borel) meager set.
Although this question is open, there are known results which answer it affirmatively in various groups or classes of groups.
For example, the following surprisingly short calculation from [26] shows that the existence of a nonempty openly Haar null set in the group implies the existence of a decomposition. (Openly Haar null sets are introduced in Subsection 4.5.)

Theorem 5.4.3 (Cohen-Kallman). If there is a nonempty openly Haar null set in G, then every countable subset C ⊆ G is contained in a comeager Haar null set. In particular, G may be written as the union G = A ∪ B where A is a Haar null set and B is meager in G.
Proof. If there is a nonempty openly Haar null set in G, then applying Lemma 3.2.10 yields that every countable set in G is openly Haar null. In particular a dense countable set C ⊇ C is openly Haar null. Then Proposition 4.5.2 yields that C ⊆ A for a G δ Haar null set A. The dense G δ set A is a countable intersection of dense open sets, hence B := G \ A is a countable union of nowhere dense sets, that is, B is (F σ and) meager. (1) Uncountable groups that admit a two-sided invariant metric.
(2) S ∞ , the group of permutations of N.  Before these, (2) had been already proved in [41] (using a different approach which is briefly described in Subsection 6.5).
The result [31, Corollary 5.2] also gives an affirmative answer in two well-known automorphism groups in addition to the already mentioned Aut(Q, ):  These results from [31] are corollaries of the examination of the 'size' of the conjugacy classes in these groups; we omit their relatively long proofs.
It is well known that Aut(B ∞ ) (the group considered in part (2) By [40,Proposition 8], the inverse image of a Haar null set under a continuous, surjective homomorphism is also Haar null, hence ϕ −1 (N ) is Haar null in G.
As M is meager, there are closed, nowhere dense sets (S n ) n∈ω such that M ⊆ n∈ω S n . For each n, the set ϕ −1 (S n ) is closed (because ϕ is continuous) and nowhere dense (because ϕ is open); these imply that ϕ −1 (M ) is meager.
Applying this, it is easy to prove the following:  (2) H(D 2 ), the group of orientation-preserving self-homeomorphisms of the closed disc D 2 . Proof.
(1) The map ϕ : (Diff k is a continuous, surjective homomorphism. As R + is locally compact, the existence of a decomposition is well known. (2) The map ψ : H(D 2 ) → H(S 1 ), h → h| S 1 is a continuous homomorphism. It is easy to see that if we restrict ψ to the set {h : h preserves the center and is linear on the radiuses} ⊂ H(D 2 ), then we get a bijective map. This shows that ψ is surjective. The other condition of Proposition 5.4.8 follows from Theorem 5.4.7 (2), which states that H(S 1 ) can be written as the union of a meager and a Haar null set.
We note that part (1) was originally proved in [26] using part (e) of [26,Proposition 2] (which we already stated without proof as part (5)  (1) H(D n ), the group of orientation-preserving self-homeomorphisms of the closed n-ball D n (where n 3 is an integer).
(2) H(S n ), the group of orientation-preserving self-homeomorphisms of the n-sphere S n (where n 2 is an integer).
After these particular cases, we would like to mention an equivalent form of Question 5.4.1, which connects it to a different family of problems. The following, seemingly unrelated question appears as [46,Question 5.5] and later in [8]: Question 5.4.11 (Elekes-Vidnyánszky, Banakh). Is each countable subset of an uncountable Polish group G contained in a G δ Haar null subset of G?
However, it turns out that this is equivalent to Question 5.4.1: Theorem 5.4.12. In an uncountable Polish group G the following are equivalent.
(1) Each countable subset of G is contained in a G δ Haar null subset of G.
(2) There exists a countable subset of G that is dense and contained in a G δ Haar null subset of G.
(3) G can be written as the union of a meager and a Haar null set. Proof.
(1) ⇒ (2): As G is Polish, there exists a countable dense subset D in G.
(2) ⇒ (3) Let D ⊂ G be countable dense set that is contained in a G δ Haar null set N ⊂ G. N is dense, because it contains the dense set D as a subset. It is well known that the complement of a dense G δ set is meager; therefore G = (G \ N ) ∪ N is a suitable decomposition.
(3) ⇒ (1): Assume that G = M 0 ∪ N 0 where M 0 is meager and N 0 is Haar null. As M 0 is meager, there exists an F σ meager set M such that M 0 ⊆ M ⊂ G (this follows from the fact that the closure of a nowhere dense set is still nowhere dense). Then the G δ set N = G \ M is Haar null, because it is a subset of N 0 .
Let C ⊂ G be an arbitrary countable set. We will show that there exists a g ∈ G such that C ⊂ gN = {g · x : x ∈ N } (this is sufficient, because the translates of a G δ Haar null set are also G δ Haar null sets). The set of 'bad' translations is which is a meager set (because it is a countable union of meager sets); this clearly shows that there exists a g ∈ G such that C ⊂ gN .
As every Haar meager set is meager (but the converse is not true in general -see  In particular, this holds for G = R ω or G = X where X is a Banach space.
The proof of this result can be found as [38,Theorem 22].

Common techniques
In this section we introduce six techniques, which are frequently useful in practice. The first five of these can be used to show that a set is small, and the last one can be used to show that a set is not small. Note that some of the results from the earlier sections (for example, the basic properties in Subsection 3.2 or the equivalent definitions in Subsection 4.1) are also very useful in practice.

Probes
Probes are a very basic technique for constructing witness measures. The core of this idea is fairly straightforward and the only surprising thing about probes is the fact that despite their simplicity they are often useful. Probes were introduced with the following definition in [62] (this paper examines the Haar null sets in completely metrizable linear spaces). There is nothing 'magical' about this definition, but it is easy to handle these simple witness measures in the calculations and if there is a probe for a set A ⊇ V , then by definition A is Haar null. In arbitrary Polish groups it is easy to generalize this idea and consider a witness measure which is the 'natural' measure supported on a small and well-understood subgroup or subset. If the considered set and the candidate for the probe are not too contrived, then it is often easy to see that it is indeed a probe.
For Haar meager sets the analogue of this is basically proving that the set is strongly Haar meager (see Subsection 4.7) and this is witnessed by a 'naturally chosen' set.
The proof of the following example illustrates the usage of probes. Fix a function ϕ ∈ C[0, 1] such that its restriction to I is not of bounded variation. We show that the one-dimensional subspace Rϕ = {c · ϕ : c ∈ R} is a probe for M (I), that is, the measure μ on C[0, 1] that is defined by (where λ is the Lebesgue measure on R) is a witness measure for M (I).
We For additional examples which demonstrate the usage of probes, see the paper [62] (which introduces probes and gives several simple examples) or the paper [61] which proves Example 6.3.1 using a two-dimensional probe. The latter paper is of particular interest because it also proves (at the beginning of Section 2) that Example 6.3.1 cannot be proved using a one-dimensional probe.

Application of the Steinhaus theorem
Sometimes the application of one of the results in Subsection 5.2 can yield very short proofs for the Haar nullness and Haar meagerness of certain sets. Unfortunately, this technique is restricted in the sense that 'good' analogs of the Steinhaus theorem are known only in special classes of groups.
We illustrate this technique by proving Example 6.1.3 again. It is well known and easy to prove that the difference of two monotone functions is a function of bounded variation; this implies that every f ∈ M (I) − M (I) satisfies that the restriction f | I is of bounded variation. However, it is easy to construct a continuous function f such that f ∞ is small and f | I is not of bounded variation; this clearly contradicts the fact that M (I) − M (I) is a neighborhood of the constant 0 function.
The proof of Proposition 6.4.3 is another example of this technique.

The Wiener measure as witness
The Wiener measure (which we will denote by μ in this section) is a well-studied Borel probability measure on the Polish group (C[0, 1], +). There are lots of results which show that something is true for 'most' continuous functions by proving that μ(E) = 0 where E is the set of 'exceptional' functions.
Although μ(E) = 0 does not necessarily imply that E is Haar null, it is often possible to use the same methods to show that there is a Borel set B such that E ⊆ B ⊆ C[0, 1] and μ(B + g) = 0 for every g ∈ C[0, 1].
We illustrate this technique on the main result of [61]. This original proof by Hunt relied on constructing a two-dimensional probe. Later, Holický and Zajíček gave an alternative proof in [60] using the Wiener measure as a witness measure; here we reproduce this second proof. Proof (Holický-Zajíček). We will prove that the set For a function f ∈ C[0, 1], we say that f is Lipschitz from the right at x if lim sup It is clearly sufficient to prove the following claim: (1) E R ⊆ B.
Here (1) is true because if f has a finite right derivative d at some point 0 x < 1, then lim sup (2) follows from the fact that B = n E n where Elementary calculations (which can be found in [78,Chapter 11]) show that E n is closed.
(3) is the nontrivial part of this claim. To prove it, first note that if f is Lipschitz from the right at x, then there exists k ∈ N and δ > 0 such that if x u v x + δ, then If n ∈ N is large enough (n > 1 5δ ), then there exists an i ∈ {0, 1, . . . , n − 1} such that i−1 n < x i n < i+1 n < i+2 n < i+3 n < x + δ, and hence for all j ∈ {0, 1, 2}, n .
If we formalize this observation, then we get Now we fix an arbitrary g ∈ C[0, 1] and conclude the proof of the claim by showing that μ(B + g) = 0.
and Δ i,j,n is the difference of two values of g (we will not use its value).
If h is a random function with the Wiener measure as its distribution, then it is well known that any difference of the form h(t 2 ) − h(t 1 ) (where t 2 > t 1 ) has normal distributions with variance t 2 − t 1 and moreover, a collection of these differences is independent if they correspond to nonoverlapping intervals.
If X is a random variable with normal distribution and variance σ 2 in an arbitrary probability space (Ω, F, P ) and Δ ∈ R and r > 0 are arbitrary values, then clearly 2πσ 2 is the maximal value of the density function of X. Using this for the differences of the values of h yields and therefore as we claimed.
Note that [89] shows that the set is not Haar null, but its complement is also not Haar null.

Compact sets are small
This technique is based on the idea that in non-locally compact groups the compact sets are 'small' in the sense that they have empty interior. This naturally inspires the following question: Question 6.4.1. Is it true that the compact subsets are Haar null (or Haar meager) in every non-locally compact Polish group? While this question is still open, there are several partial results which give positive answers when some additional assumptions are satisfied. It is frequently possible to use these partial results as lemmas. As both the system of Haar null sets and the system of Haar meager sets are σ-ideals, they also imply that K σ sets (countable unions of compact sets) are Haar null and/or Haar meager in these groups.
The following theorem is [40,Proposition 12], one of the earliest results in this topic.
Theorem 6.4.2. Let G be a non-locally compact Polish group admitting a two-sided invariant metric. Then every compact subset of G is Haar null.
Proof. We will use Theorem 4.1.7 to prove this result; this proof is not essentially different from the proof in [40], but separates the ideas specific to compact sets (this proof) and the construction of a limit measure (the proof of Theorem 4.1.7).
Fix a two-sided invariant metric d on G and let C ⊆ G be an arbitrary compact subset. We need to prove that for every δ > 0 and neighborhood U of 1 G there exists a Borel probability measure μ on G such that the support of μ is contained in U and μ(gCh) < δ for every g, h ∈ G.
Fix δ > 0 and a neighborhood U of 1 G . We may assume that U is open. As G is non-locally compact, the open set U is not totally bounded, hence there exists an ε > 0 such that U cannot be covered by finitely many open balls of radius 2ε.
As C is compact, hence totally bounded, there exists an N ∈ ω such that C can be covered by N open balls of radius ε. This means that if X ⊆ C and every x, x ∈ X satisfies x = x ⇒ d(x, x ) 2ε, then |X| N (because each of the N open balls of radius ε covering C may contain at most one element of X). Using the invariance of d this yields that for every g, h ∈ G if X ⊆ gCh and every x, x ∈ X satisfies x = x ⇒ d(x, x ) 2ε, then |X| N .
As U cannot be covered by finitely many open balls of radius 2ε, it is possible to choose a sequence (u n ) n∈ω such that u n ∈ U and u n / ∈ n−1 i=0 B(u i , 2ε) for every n ∈ ω. Choose an integer M that is larger than N δ and let Y = {u n : 0 n < M}. Let μ be the measure on Y which assigns measure 1 M to every point in Y . If g, h ∈ G are arbitrary, then μ(gCh) = |gCh∩Y | M and here every y, y ∈ gCh ∩ Y satisfies y = y ⇒ d(y, y ) 2ε, and hence |gCh ∩ Y | N , and thus μ(gCh) N M < δ.
The following result works in all non-locally compact Polish groups, but only proves that the compact sets are right Haar null (this is a weaker notion than Haar nullness, see Subsection 4.4 for the definition and properties). Proof. Suppose that C ⊆ G is compact but not right Haar null. Applying Theorem 5.2.3 yields that there exist an n ∈ ω and h 0 , h 1 , . . . , h n−1 ∈ G such that is a neighborhood of 1 G . But CC −1 is compact (as it is the image of C × C under the continuous map (x, y) → xy −1 ), thus its conjugates are also compact, and the union of finitely many compact sets is also compact, and this is a contradiction, because a neighborhood cannot be compact in G.
The paper [39] investigates the question in the case of Haar meager sets, we state the main results without proofs. This article introduces the finite translation property with the following definition: The first part of the proof is the following result which allows using this property to prove that a set is strongly Haar meager (this is a stronger notion than Haar meagerness, see Subsection 4.7 for the definition and properties). The role of this result is roughly similar to the role of Theorem 4.1.7 in the case of measure; its proof involves a relatively complex recursive construction.
Theorem 6.4.5. If an F σ set A ⊆ G has the finite translation property, then A is strongly Haar meager.
The second part is showing that the compact sets have the finite translation property when there is a two-sided invariant metric; the proof is relatively simple and very similar to the one used in Theorem 6.4.2.
Theorem 6.4.6. Let G be a non-locally compact Polish group admitting a two-sided invariant metric. Then every compact subset of G has the finite translation property.
These results yield the analogue of Theorem 6.4.2. Note that in the case when G is abelian, it is also possible to prove this as a corollary of Theorem 5.2.11, using the method of the proof of Proposition 6.4.3.
Corollary 6.4.7. Let G be a non-locally compact Polish group admitting a two-sided invariant metric. Then every compact subset of G is (strongly) Haar meager.
In addition to these, [39] also shows that compact sets have the finite translation property in S ∞ (the group of all permutations of a countably infinite set).
We illustrate the usage of this technique with a simple example.
Example 6.4.8. In the non-locally compact Polish group (Z ω , +) there are subsets A, B ⊆ Z ω such that they are neither Haar null nor Haar meager, but for every x ∈ Z ω the intersection (A + x) ∩ B is both Haar null and Haar meager.
Proof. It is well known that in Z ω a closed set C is compact if and only if C ⊆ n∈ω {u n , u n + 1, u n + 2, . . . , v n − 1, v n } for some u, v ∈ Z ω , as sets of this kind are closed subsets in a product of compact sets and in the other direction if C ⊆ Z ω is compact, then the projections map it into compact subsets of Z.
Let A = {a ∈ Z ω : a n 0 for every n ∈ ω} and B = {b ∈ Z ω : b n 0 for every n ∈ ω}. It is easy to check that these sets satisfy the condition of Lemma 6.6.2 (the result used in the last technique) and this implies that A and B are neither Haar null nor Haar meager.
On the other hand, the set (A + x) ∩ B = {z ∈ Z ω : 0 z n x n } is compact, hence Theorem 6.4.2 and Corollary 6.4.7 show that it is Haar null and Haar meager.
Note that this phenomenon is impossible in the locally compact case, where nonsmall sets have density points and if we translate a density point of one set onto a density point of the other, then the intersection will be nonsmall. On the other hand, [73,Theorem 4] proves that if (G, +) is abelian and non-locally compact, then there are sets A, B ⊂ G which are not Haar null, but satisfy that (A + x) ∩ B is Haar null for every x ∈ G.

Random construction
This is a technique that is useful when one wants to prove that a not very small set is still small enough to be Haar null. (For example, this technique may work for sets that are not Haar meager, because it does not prove Haar meagerness.) The main idea of this technique is that a witness measure for a Haar null set A ⊆ G is a Borel probability measure and one can use the language of probability theory (for example, random variables, conditional probabilities, and stochastic processes) to construct it and prove that it is indeed a witness measure.
For example, the paper [41] applies this to prove a result in S ∞ , the group of all permutations of the natural numbers (endowed with the topology of pointwise convergence). We illustrate this technique by reproducing the core ideas of this proof. (The proof also contains relatively long calculations which we omit.) Theorem 6.5.1 (Dougherty-Mycielski). In the Polish group S ∞ , let X be the set of permutations that have infinitely many infinite cycles and finitely many finite cycles. Then the complement of X is Haar null.
Proof (sketch). We will find a Borel probability measure μ on S ∞ such that μ(gXh) = 1 for every g, h ∈ S ∞ . Using that X is conjugacy invariant μ(gXh) = μ(gh(h −1 Xh)) = μ(ghX) and We define the probability measure μ by describing a procedure which chooses a random permutation p with distribution μ. (This way we can describe a relatively complicated measure in a way that keeps the calculations manageable.) Fix a sequence k 0 < k 1 < k 2 < · · · of natural numbers which are large enough (the actual growth rate is used by the omitted parts of the proof). The procedure will choose values for p(0), p −1 (0), p(1), p −1 (1), p(2), p −1 (2), . . . in this order, skipping those which are already defined (for example, if we choose p(0) = 1 in the first step, then the step for p −1 (1) is omitted, as we already know that p −1 (1) = 0). When we have to choose a value for p(n), we choose randomly a natural number n < k n which is still available as an image (that is, n n and n is not among the already determined values p(0), p(1), . . . , p(n − 1)); we assign equal probabilities to each of these choices. Similarly, when we have to choose a value for p −1 (n), we choose randomly a natural number n k n which is still available as a preimage (that is, n > n and n is not among the already determined values p −1 (0), p −1 (1), . . . , p −1 (n − 1)); we assign equal probabilities to each of these choices again. When we are finished with these steps, the resulting object p is clearly a well-defined permutation, as every n ∈ ω has exactly one image and exactly one preimage assigned to it. We can assume that k n is large enough to satisfy k n > 2n + 1 and this guarantees that we never 'run out' of choices.
We say that p 0 is a possible partial result, if it can arise after finitely many steps of this process. We omit the relatively long combinatorial arguments which show that the following claim is true: Claim 6.5.2. Assume that p 0 is a possible partial result, g ∈ S ∞ is an arbitrary element, and M is a natural number. Then there is a natural number N such that the conditional probability with respect to μ, under the condition of extending p 0 , of the event that the permutation p chosen by our process will be such that gp has no finite cycles including a number greater than N and no two of the numbers N + 1, . . . , N + M are in the same cycle of gp is at least 1 2 .
Using this claim, it is possible to show the following claim by induction on i (we also omit this part of the proof): Claim 6.5.3. Assume that (as in the previous claim) p 0 is a possible partial result, g ∈ S ∞ is an arbitrary element, and M is a natural number. Then for every i ∈ ω the conditional probability (with respect to μ, under the condition of extending p 0 ) of the event that the permutation p chosen by our process satisfies that gp has only finitely many finite cycles and at least M infinite cycles is at least 1 − 2 −i .
Applying this second claim for every i ∈ ω in the special case when p 0 is the empty partial permutation yields that the (unconditional) probability (with respect to μ) of the event that the permutation p chosen by our process satisfies that gp has only finitely many finite cycles and at least M infinite cycles is 1. Since this is true for every M ∈ ω, the permutation gp has infinitely many infinite cycles with μ-probability 1. This shows that μ(g −1 X) = 1 for the arbitrary permutation g, so we are done.
This set X is the union of countably many conjugacy classes of permutations, one for each finite list of sizes for the finite cycles in the permutation; the paper [41] also shows that none of these conjugacy classes are Haar null.

Sets containing translates of all compact sets
Proving that a set is not Haar null from the definitions requires showing that all Borel probability measures fail to witness that it is Haar null, which is frequently harder than just showing one measure witnessing that the set is Haar null. The situation is similar for Haar meager sets, where even the choice of the domain of the witness function is not straightforward, although the equivalence (1) ⇔ (2) in Theorem 4.1.11 can be used to eliminate this additional choice.
However, there are many cases where it is sufficient to apply the following notion and lemma: Definition 6.6.1. We say that a set A ⊆ G is compact catcher if every compact set can be translated into it (that is, for every compact set C ⊆ G there are g, h ∈ G such that gCh ⊆ A).
Note that these sets are called 'thick sets' in [9, Definition 7.1]. Lemma 6.6.2. If A ⊆ G is compact catcher, then it is neither generalized Haar null nor Haar meager. This lemma appears, for example, as [87, Lemma 2.1], but there are many papers where this simple idea is integrated into the proof of some result.
Proof. If A were generalized Haar null, then by Theorem 4.1.4 there would be a universally measurable set B ⊇ A and a Borel probability measure μ with compact support C ⊆ G such that μ(g Bh ) = 0 for every g , h ∈ G, but there are g, h ∈ G such that gCh ⊆ A ⊆ B and thus μ(g −1 Bh −1 ) μ(C) = 1, a contradiction.
Similarly, if A would be Haar meager, then there would be a Borel set B ⊇ A, a (nonempty) compact metric space K, and a continuous function f : K → G such that f −1 (g Bh ) is meager in K for every g , h ∈ G, but there are g, h ∈ G such that gf (K)h ⊆ A ⊆ B and thus f −1 (g −1 Bh −1 ) ⊇ f −1 (f (K)) = K is not meager in K, a contradiction.

A brief outlook
It is clearly beyond the scope of this paper to collect the countless number of results and applications of Haar null sets in various fields of mathematics. However, we now give a highly incomplete list of works using Haar null sets, and encourage the reader to use these as starting points for further reading.
First we mention the paper [77] which also surveys applications of Haar null sets in addition to presenting their core properties. However, this paper was published in 2005 and therefore it does not contain several recent ideas.
The recent paper [9] introduces and systematically investigates (in the abelian case) the socalled Haar-I sets, a common generalization of the notions of Haar null and Haar meager sets.
One of the original motivations of Christensen was to consider versions of automatic continuity as discussed, for example, in Corollary 5.2.5, in this topic see the excellent survey paper [81], and also [52] and [82].
The Rademacher theorem states that a Lipschitz function between Euclidean spaces is differentiable almost everywhere. The second motivation of Christensen was to extend this result to Banach spaces, see the paper [24] by Christensen, and also [15][16][17][18]90].
A closely related result is the Alexandrov theorem stating that convex functions on Euclidean spaces are twice differentiable almost everywhere. For extensions of this result to Banach spaces see, for example, [71,72].
There are various other interesting directions of research in functional analysis involving Haar null sets, see, for example, [42,49,69,70].
When Haar null sets were rediscovered (under the name 'shy sets') in [62], they were the first ones to apply Haar null sets in the theory of dynamical systems. Since then numerous other such applications have been found, see, for example, [2,14,19,66].
Multifractal analysis is a large area within geometric measure theory. There are numerous interesting results concerning the multifractal analysis of co-Haar null functions and measures, see, for example, [1,10,12,76].
There are countless results in geometric measure theory calculating various fractal dimensions of images, graphs, and level sets of certain functions, for example, the generic ones, co-Haar null ones, and Brownian motion. A recurring theme is that the generic set is as small as it can, be, while the co-Haar null one is a large as it can be. See, for example, [3, 4, 6, 11, 50, 53, 54, 57, 65, 74].
Let us also mention a few results that are measure theoretic duals of classical results of real analysis about generic continuous functions: [60,61,89] concern nowhere differentiable functions, and [5] is the Bruckner-Garg theorem describing the topological structure of the level sets. Interestingly, in the case of [5] and [89] the generic behavior happens with 'probability strictly between 0 and 1', that is, the set of functions exhibiting the behavior in question is neither Haar null nor co-Haar null.
A large part of modern descriptive set theory deals with homeomorphism groups of compact metric spaces, and automorphism groups of countable first-order structures, such as H[0, 1], the group on increasing homeomorphisms of the unit interval, S ∞ , the permutation group of the natural numbers or Aut(Q, <), and the group of increasing bijections of the rational numbers. There have been numerous papers describing the structure of the random element of these groups (that is, the properties which are true for all elements except for a Haar null set), see, for example, [26, 30-33, 41, 87].
There is also quite some literature dealing with the set theoretic aspects of Haar null sets. The so-called cardinal invariants of (various versions of) Haar null sets are calculated in [7] and [44]; other related results can be found in, for example, [84] and [45].
Finally, there are also interesting but somewhat sporadic results involving Haar null sets in the theory of differential equations: [48,64] and in the theory of functional equations: [20,25,55,62,80].

List of open questions
This section collects the open questions which are mentioned in this survey. As this survey is limited to studying fundamental properties and useful techniques, this collection is also limited to open questions from these topics, and we do not mention an enormous number of problems concerning other aspects of Haar null and Haar meager sets. Most of these questions outside the scope of our survey can be found in the references mentioned in Section 7. The following question is the main problem of Subsection 5.2, where we describe lots of partial results (there are groups where the answer is negative, but some weaker variants can be answered positively): Question 5.2.2. Let A be a Borel (or universally measurable) subset in the Polish group G that is not Haar null (or not generalized Haar null, or not Haar meager). What can we say about the groups G where 1 G ∈ int(AA −1 ) is neccessarily satisfied?
The following question is the main problem of Subsection 5.4, where we describe lots of positive partial results: Question 5.4.1 [28]. Can every uncountable Polish group be written as the union of two sets, one meager and the other Haar null?
In Theorem 5.4.12 we prove that the following question from [8; 46, Question 5.5] is equivalent to the previously stated one: Question 5.4.11 (Elekes-Vindnyánszky, Banakh). Is each countable subset of an uncountable Polish group G contained in a G δ Haar null subset of G?
The questions [33, Question 6.6] and [33, Question 6.7] highlight the following special cases of Question 5.4.1: Question 5.4.10 (Darji-Elekes-Kalina-Kiss-Vidnyánszky). Is it possible to write the following uncountable Polish groups as the union of a meager and a Haar null set?
(1) H(D n ), the group of orientation-preserving self-homeomorphisms of the closed n-ball D n (where n 3 is an integer). (2) H(S n ), the group of orientation-preserving self-homeomorphisms of the n-sphere S n (where n 2 is an integer). As every Haar meager set is meager (but the converse is not true in general -see Subsection 3.3), the following question is a stronger variant of Question 5.4.1: Question 5.4.13 [63,Question 4]. Can every uncountable abelian Polish group be written as the union of two sets, one Haar meager and the other Haar null?
The paper [38] answers this question positively in some classes of groups. In non-locally compact groups the compact sets are 'small' in the sense that they have empty interior. This inspires the following question: Question 6.4.1. Is it true that the compact subsets are Haar null (or Haar meager) in every non-locally compact Polish group?
Several papers contain partial answers to this question, these are summarized in Subsection 6.4.
Let us conclude this list by asking the following vague question.
Question 8.1. Can one prove some sort of uniqueness of the ideal of Haar null sets? Somewhat more precisely, can one prove that if I is a σ-ideal (say on Z ω ) that is translationinvariant, has a Borel basis, contains the compact sets, is orthogonal to M, definable in some sense, and possesses certain further 'natural' properties, then I = HN ?
Interesting results of very similar nature can be found in [51,91], but these works build heavily on properties of Lebesgue-null sets not shared by Haar null sets in non-locally compact groups, for example, ccc-ness and Fubini properties.