The celebrated theorem of Halmos and Savage implies that if M is a
set of $\mathbb{P}$-absolutely continuous probability measures Q on $(\Omega,
F, \mathbb{P})$ such that each $A \in F, \mathbb{P}(A) > 0$ is charged by
some $Q\in M$, 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 $Q_0 \in M$ with full support; that is,
$\mathbb{P}(A) > 0$ implies $Q_0(A) > 0 $. We show a quantitative
version: if we assume that, for $\varepsilon > 0$ and $\delta > 0$ fixed,
$\mathbb{P}(A)> \varepsilon$ implies that there is $Q \in M$ and $Q(A) >
\delta$, then there is $Q_0 \in M$ such that $\mathbb{P}(A) >4 \varepsilon$
implies $Q_0(A)>\varepsilon^2 \delta/2$. This version of the Halmos-Savage
theorem also allows a "dualization" which we also prove in a
quantitative and qualitative version. We give applications to asymptoic
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.