Undecidably semilocalizable metric measure spaces

TL;DR

This paper characterizes measure spaces where the canonical map from $L_0$ to the dual of $L_$ is surjective, providing conditions that are sometimes undecidable within standard set theory.

Contribution

It introduces two equivalent conditions for measure spaces related to $L_0$ and $L_^*$ surjectivity, including cases with undecidable properties in ZFC.

Findings

Conditions for surjectivity depend on order completeness and space decomposition.

Examples show some properties are undecidable in ZFC.

Includes cases where Hausdorff dimension influences decidability.

Abstract

We characterize measure spaces such that the canonical map $L_\infty \to L_1^*$ is surjective. In case of $d$ dimensional Hausdorff measure of a complete separable metric space $X$ we give two equivalent conditions. One is in terms of the order completeness of a quotient Boolean algebra associated with measurable sets and with locally null sets. Another one is in terms of the possibility to decompose space in a certain way into sets of nonzero finite measure. We give examples of $X$ and $d$ so that whether these conditions are met is undecidable in ZFC, including one with $d$ equals the Hausdorff dimension of $X$.