Extension of measures: a categorical approach

Abstract

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.