On the Existence of a Minimal Sufficient Subfield

Abstract

Let $(X, S)$ be a measurable space and $M$ a collection of probability measures on S. Pitcher (Pacific J. Math. (1965), pages 597-611) introduced a condition called compactness on the statistical structure $(X, S, M)$, more general than domination by a fixed $\sigma$-finite measure. Under this condition he gave a construction of a minimal sufficient subfield. In this paper a counterexample invalidating this construction is presented. We give a nonconstructive proof of the existence of a minimal sufficient subfield under a condition slightly weaker than compactness. The proof proceeds by considering the intersection of an uncountable collection of sufficient subfields, and relies on a martingale convergence theorem with directed index set, due to Krickeberg.