Measurable tail disintegrations of the Haar integral are purely finitely additive

Abstract:

There are countably additive probability measures, P, and sub-sigma fields, relative to which P admits no proper, measurable, conditional distributions, except, possibly, those which are purely finitely additive. The usual fair, coin-tossing probability measure and the tail sigma field illustrate this phenomenon. More generally, every measurable, disintegration of the Haar integral of any compact metrizable group, G, relative to the partition, $\Pi$, of G which consists of the left cosets of any dense denumerable subgroup S of G, or what comes to the same thing, relative to the sigma field of Haar-measurable subsets of G which are invariant under right translation by S, is purely finitely additive.