A note on $\sigma$-algebras on sets of affine and measurable maps to the unit interval

TL;DR

This paper examines two different $\sigma$-algebras on convex spaces of functions to the unit interval, providing counterexamples that challenge previous assertions about their equivalence.

Contribution

It offers explicit examples showing that the two $\sigma$-algebras do not always coincide, contradicting earlier claims in the literature.

Findings

Counterexamples demonstrate the $\sigma$-algebras can differ

Challenges previous assumptions about their equivalence

Highlights the need for careful analysis in measure-theoretic structures

Abstract

In The factorization of the Giry monad (arXiv:1707.00488v2) the author considers two $\sigma$-algebras on convex spaces of functions to the unit interval. One of them is generated by the Boolean subobjects and the other is the $\sigma$-algebra induced by the evaluation maps. The author asserts that, under the assumptions given in the paper, the two $\sigma$-algebras coincide. We give examples contradicting this statement.