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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 …

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 answer to the first part is no. The point is that surjectivity is not as strong a condition in this context as one might wish for. Let $X=Y=Z=[0,1]$ with the Borel $\sigma$-algebra and $\mu_X=\mu_ …

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 …

To close a gap: From the answer of Gerald Edgar, we know that the answer to the second question is no if the spaces involved have cardinality larger than $\mathfrak{c}$. This leaves open what happens …

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 …

The problem of choosing subsets at random has been studied in a rather different context in mathematical economics. Suppose we choose a subset of $[0,1]$ by independently throwing a fair coin for each …