On a Lusin theorem for capacities

TL;DR

This paper establishes a condition under which Lusin's theorem applies to capacities on compact metric spaces, specifically that the capacity must be continuous from above.

Contribution

It characterizes when Lusin's theorem holds for capacities, linking it to the property of being continuous from above.

Findings

Lusin's theorem holds for a capacity if and only if the capacity is continuous from above.

Provides a necessary and sufficient condition for Lusin's theorem in the context of capacities.

Connects measure-theoretic properties with topological properties of capacities.

Abstract

Let $X$ be a compact metric space and let $v$ be a sub-additive capacity defined on $X$. We show that Lusin's theorem with respect to $v$ holds if and only if $v$ is continuous from above.