The celebrated theorem of Halmos and Savage implies that if M is a set of BP-absolutely continuous probability measures Q on (Omega,F,P) such that each A is an element of F, P(A) > 0 is charged by some Q is an element of M, that is, Q(A) > 0 (where the choice of Q depends on the set A), then-provided M is closed under countable convex combinations-we can find Q(0) is an element of M with full support; that is, P(A) > 0 implies Q(0)(A) > 0. We show a quantitative version: if we assume that, for epsilon > 0 and delta > 0 fixed, P(A) > is an element of implies that there is Q is an element of M and Q(A) > delta, then there is Q(0) is an element of M such that P(A) > 4 epsilon implies Q(0)(A) > epsilon(2) delta/2. This version of the Halmos-Savage theorem also allows a ''dualization'' which we also prove in a quantitative and a qualitative version. We give applications to asymptotic problems arising in mathematical finance and pertaining to the relation of the concept of ''no arbitrage'' and the existence of equivalent martingale measures for a sequence of stochastic processes.
