CH, V=L, Disintegrations of Measures, and {\Pi}^1_1 Sets

TL;DR

This paper explores the set-theoretic independence of Maharam's question on measure disintegrations, showing that the answer varies with different axioms like CH and V=L, and constructs special a1^1_1 sets.

Contribution

It demonstrates the independence of Maharam's question from ZFC and constructs a1^1_1 sets with unique properties under certain axioms.

Findings

Under CH, the answer to Maharam's question is no.

Under V=L, a1^1_1 sets with special properties can be constructed.

The results connect measure theory with infinitary combinatorics and set-theoretic axioms.

Abstract

In 1950 Maharam asked whether every disintegration of a $\sigma$-finite measure into $\sigma$-finite measures is necessarily uniformly $\sigma$-finite. Over the years under special conditions on the disintegration, the answer was shown to be yes. However, we show here that the answer may depend on the axioms of set theory in the following sense. If CH, the continuum hypothesis holds, then the answer is no. One proof of this leads to some interesting problems in infinitary combinatorics. If G\"odel's axiom of constructibility $\mathbf{V}=\mathbf{L}$ holds, then not only is the answer no, but, of equal interest is the construction of $\mathbf{\Pi}^1_1$ sets with very special properties.