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Original Paper
A localic theory of lower and upper integrals
Article first published online: 30 JAN 2008
DOI: 10.1002/malq.200710028
Copyright © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Vickers, S. (2008), A localic theory of lower and upper integrals. Mathematical Logic Quarterly, 54: 109–123. doi: 10.1002/malq.200710028
Publication History
- Issue published online: 30 JAN 2008
- Article first published online: 30 JAN 2008
- Manuscript Accepted: 3 JUL 2007
- Manuscript Revised: 6 JUN 2007
- Manuscript Received: 15 APR 2007
- Abstract
- References
- Cited By
Keywords:
- Riemann integral;
- Choquet integral;
- valuation;
- locale;
- geometric logic
Abstract
An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.
Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)