Undecidably semilocalizable metric measure spaces

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 $X$.