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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?

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    I think it's probably true if one of the inequalities is weak.2012-05-18
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    On a finite space probability space you use the hahn-banach thm or stiemke alternative thm, but the thing that separates S from $\nu$ wants to be in the dual of measures2012-05-18
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    What exactly do you mean by "mixture" in this context?2012-05-18
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    See my other question on stackexchange [here](http://math.stackexchange.com/questions/141744/generalized-notions-of-mixture).2012-05-18
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    "Mixture" should certainly include weighted averages of finitely many of these probability measures, and I would think weak limits of them. But it doesn't look as if we've got enough structure on the probability space to define such a concept as weak limits.2012-05-18
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    Ok, but for guy's answer to apply, you have to settle some measurability issues. In order to make sense of $P(A) = \int_\mathcal{M} P_\mu(A) Q(d\mu)$, you need at a minimum that for each $A$, the map $\mu \mapsto P_\mu(A)$ is measurable (with respect to some underlying $\sigma$-algebra on $\mathcal{M}$). I suspect this will not be enough to prove your result. Having weak convergence available would be better, but you would need a topology on your measure space.2012-05-18
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    I don't understand. I was pretty satisfied with the answer I got to my previous question. Also, if I understand correctly what a weak limit of probability measures is, it doesn't require any additional structure to be defined, but a weak limit of members of $S$ should count as a mixture of members of $S$. (Maybe it is a mixture by a merely finitely additive mixing measure?) Please be patient since this is not my field.2012-05-18
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    The last comment I wrote was actually a response to Michael. I only just now am seeing Nate's comment. Nate, I was thinking the mixtures would be with respect to the weakest $\sigma$-algebra necessary for the maps you mention to be measurable. That will give us the most generous possible conception of mixture, right? (But I fear I am missing something important.)2012-05-18

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