Suppose I have a probability measure $\nu$ and a set of probability measures $S$ (all defined on the same $\sigma$-algebra). Are the following two statements equivalent?
(1) $\nu$ is not a mixture of the elements of $S$.
(2) There is a random variable $X$ such that the expectation of $X$ under $\nu$ is less than 0, and the expectation of $X$ under all of the members of $S$ is greater than 0.
If not, is something similar true, or true in a special case?
Is the situation the same for merely finitely additive probability measures?