Timeline for The Borel $\sigma$-algebra of the set of probability measures

8 events

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Jan 16, 2021 at 15:59	history	edited	YCor	CC BY-SA 4.0	formatting
May 25, 2014 at 11:08	comment	added	Michael Greinecker		You can find a proof in Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve, where this is Proposition 7.25.
May 22, 2014 at 18:24	vote	accept	TV2323
May 22, 2014 at 1:11	answer	added	Nate Eldredge		timeline score: 6
May 22, 2014 at 0:51	answer	added	Alexander Shamov		timeline score: 6
May 22, 2014 at 0:49	comment	added	Nate Eldredge		To show that $\mu \mapsto \mu(A)$ is measurable for every Borel $A$, try a $\pi$-$\lambda$ or monotone class argument. Let $\mathcal{L}$ be the set of all $A \subset X$ such that $\mu \mapsto \mu(A)$ is measurable, and let $\mathcal{P}$ be the closed subsets of $X$. Then it is easy to check that $\mathcal{P}$ is closed under finite intersections, $\mathcal{L}$ is a $\lambda$-system, and $\mathcal{P} \subset \mathcal{L}$. The conclusion is that $\mathcal{L} \supset \sigma(\mathcal{P})$, i.e. the Borel sets.
May 21, 2014 at 21:52	review	First posts
May 21, 2014 at 23:14
May 21, 2014 at 21:34	history	asked	TV2323	CC BY-SA 3.0