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Here is an answer for $X$ metrizable and $\sigma$-compact. I assume $I$ is finite on compactly supported continuous functions. Recall that a locally compact metrizable $\sigma$-compact space is Poli …

It is generally true that if $(X,\mathcal{X})$ is a measurable space, $(Y,\mathcal{Y})$ a standard Borel space, and $\kappa:X\to\mathcal{P}(X)$ a transition probability, then the measure-valued funct …

There is an impossibility theorem: If you let $\mathcal{L}$ be the the space of Borel-measurable functions $f:[0,1]\to[0,1]$, and $e:\mathcal{L}\times [0,1]\to[0,1]$ the evaluation given by $e(f,x)\ma …

Minimal generators of $\sigma$-algebras are treated in Bhaskara-Rao, K. P. S., & Rao, B. V. (1981). Borel spaces. PWN. Among other things, it is shown there that every countably generated $\sig …

As Martin Sleziak has pointed out in a comment, one could only a get a pseudometric space this way and has to quotient out the null sets to obtain a proper metric. The question of separability is the …

Fix a finite measure $\mu$ on $\mathbb{R}^\infty$. For each $n\in\mathbb{N}$, there are at most countably many $x\in\mathbb{R}$ such that $$\mu\big(\mathbb{R}^{n-1}\times\{x\}\times\mathbb{R}\times\ma …

That every sequence congerging in measure has an almust surely congerging subsequence was apparently first shown by Riesz in 1909 in "Sur les suites de fonctions mesurables". I don't know about the ot …

This is the category of probabilistic mappings, introduced by Lawvere in some never published notes from 1962. This paper by Culbertson and Sturtz has references to most of the subsequent literature a …

This answer is identical to the one I gave when the same question was posted at M.SE.: There are a number of constructions that do not work for Polish spaces, but a certain class of probability space …

Sierpinski gave an example of a nonmeasurable subset of $\mathbb{R}^2$ such that all sections are singletons and hence null sets. So the answer is in general no.

One sufficient condition is that the source space is nonatomic and the target space has the Borel sets of a Polish space as the underlying $\sigma$-algebra. See here for pointers on how one can prove …

Let $F:\Omega\to 2^X$ be a measurable multifunction with closed values in the separable Banach space $X$. It seems to be implicitly assumed that $F$ jas nonempty values. $F$ is integrably bounded if …

I've found a rather simple solution. No additional assumptions are necessary. There is a natural method of composing transition probabilities that gives rise to a new transition probability (see for e …

Yes, and you can even do the approximation by a single set. All the structure you need is that $t$ is a measurable function between Polish spaces. The function $t$ is continuous and hence measurable, …

This seems to be a pure general topology problem. Enriching the topology will necessarily destroy the Prohorov property whenever the enrichment matters. If you have two nested Hausdorff topologies and …