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Title: Extension of measures: a categorical approach (English)
Author: Frič, Roman
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 130
Issue: 4
Year: 2005
Pages: 397-407
Summary lang: English
Category: math
Summary: We present a categorical approach to the extension of probabilities, i.e. normed $\sigma $-additive measures. J. Novák showed that each bounded $\sigma $-additive measure on a ring of sets $\mathbb{A}$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $\mathbb{A}$ over the generated $\sigma $-ring $\sigma (\mathbb{A})$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous functions on $X$ over its Hewitt realcompactification $\upsilon X$). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that $\sigma (\mathbb{A})$ is the sequential envelope of $\mathbb{A}$ with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category $\mathop {{\mathrm ID}}$ of $-posets of fuzzy sets (such $-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on $\mathbb{A}$ over $\sigma (\mathbb{A})$ is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems. (English)
Keyword: extension of measure
Keyword: categorical methods
Keyword: sequential continuity
Keyword: sequential envelope
Keyword: field of subsets
Keyword: D-poset of fuzzy sets
Keyword: effect algebra
Keyword: epireflection
MSC: 18B30
MSC: 28A05
MSC: 28A12
MSC: 28E10
MSC: 54A40
MSC: 54B30
MSC: 54C20
MSC: 60B99
idZBL: Zbl 1107.54014
idMR: MR2182385
Date available: 2009-09-24T22:22:48Z
Last updated: 2015-11-01
Stable URL:
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