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If $(\Omega,\Sigma_\Omega)$ is standard Borel, $E$ metrizable, and $X:\Omega_\Sigma\to E$ measurable, then $X(\Omega)$ is separable. This is Proposition 1.11 in "Probability Distributions on Banach Sp …

Here is a proof that probability measures have closed range under DC, summarized from the discussion with Iosif Pinelis. I try to be as explicit as possible about any choice arguments used. For any pr …

The assumption that $Y$ is countably separated cannot be meaningfully weakened. The following is Proposition 2.1 of [Musial, Kazimierz. "Projective limits of perfect measure spaces." Fund. Math 110.16 …

No. Let $N\subseteq[0,1]$ be a non-measurable set. Let $\mathcal{F}$ be the family of indicator functions of finite subsets of $N$. Then $\mathcal{F}$ is a net under the pointwise ordering with limit …

Graph measurability of $P$ is not sufficient. Let $E\subseteq[0,1]^2$ be a Borel set whose projection $\pi(E)$ onto the first coordinate is not Borel. Let $X=\mathbb{R}$ and let $B$ have the constant …

Let all factor spaces be nontrivial compact Hausdorff spaces. Then every continuous function is determined by countably many coordinates, and so is, consequently, every Baire measurable set. It follow …

Note: This answer used to be a counterexample that missed the mark. The way to get around he definitional issues with conditional expectations is to work with regular conditional probabilities in prod …

Yes, that is the approach of Daniell and Stone. To see how the approach works and how general it is, you can take a look at the four-part series on "Notes on Integration" by M. H. Stone. See here, her …

Essentially yes. Only essentially, because the conditional probability on the right hand has to be defined in the first place. The limit exists almost everywhere in the topology of weak convergence of …

The Borel $\sigma$-algebra on the space $C(K,E)$ of continuous functions from a compact metrizable space $K$ to a separable metric space with the induced uniform metric is generated by the evaluation …

No to both. Let $\gamma$ be the uniform distribution on $\Delta=\{(x,x)\mid x\in [0,1]\}$, the diagonal of $[0,1]^2$. The marginals are simply the uniform distribution on $[0,1]$. Fix some function $g …

Here is a positive answer for the case that $\Sigma_0$ is generated by a random variable with values in a Polish space, so that we can use regular conditional probabilities and for some kernel $\kappa …

Yes. Every uncountable Polish space is isomorphic as a measurable space to the unit interval by Kuratowski's isomorphism theorem and admits, therefore, a nonatomic probability measure. On a Polish spa …

Clearly, the integral of this correspondence coincides with the integral of correspondence $\phi:X\to 2^\mathbb{R}$ given by $\phi(x)=\big\{\langle a,v(x)\rangle \mid a\in A\big\}$. Moreover, $\phi$ …

The answer to the first question is yes. There is a class of probability spaces known under various names such as superatomless, saturated, nowhere countably-generated, $\aleph_1$-atomless, and a coup …