# Article

 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: http://hdl.handle.net/10338.dmlcz/134212 . Reference: [1] Adámek, J.: Theory of Mathematical Structures.Reidel, Dordrecht, 1983. MR 0735079 Reference: [2] Bugajski, S.: Statistical maps I. Basic properties.Math. Slovaca 51 (2001), 321–342. Zbl 1088.81021, MR 1842320 Reference: [3] Bugajski, S.: Statistical maps II. Operational random variables.Math. Slovaca 51 (2001), 343–361. 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