Extension of measures: a categorical approach

Abstract: We present a categorical approach to the extension of probabilities, i.e. normed $\sigma$-additive measures. J. Novak 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 Cech-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 $\ID$ of $\D$-posets of fuzzy sets (such $\D$-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.

Keywords: extension of measure, categorical methods, sequential continuity, sequential envelope, field of subsets, D-poset of fuzzy sets, effect algebra, epireflection

Classification (MSC2000): 54C20, 54B30, 28A12, 28E10, 28A05, 60B99

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