Disintegration of Measures , and Π 11 Sets

In 1950 Maharam asked whether every disintegration of a σ-finite measure into σ-finite measures is necessarily uniformly σ-finite. Over the years under special conditions on the disintegration, the answer was shown to be yes. We show here that the question is equivalent to the existence of a Borel uniformization of a certain set defined from the disintegration. Moreover, we show 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. Our proof of this leads to some interesting problems in infinitary combinatorics. Also, if Gödel’s axiom of constructibility V = L holds, then not only is the answer no, but, of equal interest is the construction of Π1 sets with very special properties.