Problem with almost everywhere equality

TL;DR

This paper characterizes metrizable spaces with the almost everywhere equality property, showing they are precisely those with cardinality at most continuum, linking topological and measure-theoretic properties.

Contribution

It establishes a necessary and sufficient condition for metrizable spaces to have (AEEP), connecting topological size with measure-theoretic measurability of equality sets.

Findings

Metrizable spaces have (AEEP) iff their cardinality ≤ continuum.

Spaces with larger cardinality do not have (AEEP).

Provides a characterization linking topology and measure theory.

Abstract

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$ are two measurable functions, then the set $\Delta(f,g) = \{x \in X:\ f(x) = g(x)\}$ is a member of $\mathfrak{M}$. It is shown that a metrizable space $Y$ has (AEEP) iff the cardinality of $Y$ is no greater than $2^{\aleph_0}$.