Extensions of measures and the von Neumann selection theorem

Abstract

Let (X, Bx) be a Blackwell space, where BX is the a-algebra of Borel sets. Then if a is a finite measure defined on a countably generated sub-a-algebra BG Bx, a can be extended to a Borel measure T. Equivalently, if X and Y are Blackwell andf: X-* Y is Borel, and It is a Borel measure carried on f(X)c Y, then there exists a Borel measure T on X with Tf =o, where T'(E)=T(f1 (E)). We characterize {TI Tf = a} if f is semischlicht. Let BX denote the Borel sets of a topological space X. We consider the following measure extension (or equivalently restriction) problem: given a measure (we will always mean finite measure) a defined on a or-algebra Bc Bx, can or be extended to all of Bx, i.e., does there exist a Borel measure isuch that i-(E)=a(E) for all E e B? It is well known (see [1, p. 71], for details) that if B1 and B2 are c-algebras, and B2 is generated by B1 and finitely many additional sets, then any measure on B1 can be extended to B2. The result is not known for countably generated extensions. We show below (Theorem 5) that if X is a Blackwell space and B is a countably generated sub-a-algebra of Bx, then any measure on B extends to Bx. A Blackwell space is a measure space (X, Bx), where X is an analytic subset of a complete separable metric space (c.s.m.). A subset A of a c.s.m. is analytic iff A is the continuous image of a c.s.m. We note that the analytic sets form a proper subset of Ux, the set of absolutely measurable subsets of X, where E e Ux iff E is f-measurable for all finite Borel measures 4u, where '7 denotes the completion of It, i.e., given j, there exist El, E2-e Bx such that El c Ec E2 and M(E2-E1)=O. A function g is said to be absolutely measurable if g-1(V) E Ux for all open V. Details may be found in [3], [4], or [5]. We note that if Xc S, X analytic, S a c.s.m., then Bx={Er)XjEceBS}, so elements of BX are topologically analytic, and not necessarily Borel in S. We begin by considering a special class of sub-cr-algebras of Bx. Let f:X-Y be Borel measurable, and let Bf={f-l(E)JE e Bx}. Given a Borel Presented to the Society, April 20, 1973; received by the editors January 30, 1973 and, in revised form, June 4, 1973. AMS (MOS) subject classifications (1970). Primary 28A05, 28A60, 28-00. 1 Partially supported by NSF GP-38265. ? American Mathematical Society 1974