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I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\,\mathcal{F}\right)$ be the corresponding set of probability measures.

Suppose we endow $\triangle\left(X,\,\mathcal{F}\right)$ itself with the sigma-algebra generated by sets of the form

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ A\left(E,\, p\right)=\left\{ {\mu\in\triangle\left(X,\,\mathcal{F}\right)\,|\,\mu\left(E\right)\geq p}\right\} ,\,\,\,\,\,\,\,\,\,E\in F,\,\,\ p\in[0,\,1]$

Then the above sigma-algebra coincides with the Borel sigma-algebra generated by the weak * topology.

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  • $\begingroup$ Basic open sets are defined by conditions such as $\int f\, d\mu > a$ for fixed continuous $f$. It should be routine to check (at least if it's true) that these sets can be manufactured from your $A(E,p)$'s and vice versa. $\endgroup$
    – Christian Remling
    Jan 28, 2015 at 3:13
  • $\begingroup$ Also posted on MSE (which might be the better place for the question): math.stackexchange.com/questions/1122089/… $\endgroup$
    – Christian Remling
    Jan 28, 2015 at 3:20
  • $\begingroup$ @ChristianRemling Would that not be equivalent to showing that the $A\left(E,\, p\right)$'s generate the weak * topology? The statement that I've seen is that the sigma-algebra generated directly from the $A\left(E,\, p\right)$'s, and not from the topology they generate, is what coincides with the weak * Borel algebra. $\endgroup$
    – Mark
    Jan 28, 2015 at 3:20
  • $\begingroup$ @Mark: The $A$'s don't look open to me, but one half of what I need to show is that $\sigma(A(E,p))$ contains all (weak $*$ relatively) open sets. $\endgroup$
    – Christian Remling
    Jan 28, 2015 at 3:23
  • $\begingroup$ @ChristianRemling So, it can be shown that for each $E\in F$, the function $\mu\mapsto\mu\left(E\right)$ is measurable. (Aliprantis 1999) This immediately implies that the sigma algebra generated by the $A\left(E,\, p\right)$ is a subset of the Borel sigma-algebra. For the converse, it can be shown that basis elements in the weak topology can be described as sets of the form $A\left(E,\, p\right)$, but then the result doesn't follow because sigma-algebras are only closed under countable unions. $\endgroup$
    – Mark
    Jan 29, 2015 at 2:07

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