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Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions. Is there a direct ch …

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best …

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural …

I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand. Can you provide some examples …

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\sig …

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; someti …

In addition to the excellent answers above, I also suggest the nice survey Oxtoby, Ergodic Sets (Zbl 0046.11504, MR47262, DOI: 10.1090/S0002-9904-1952-09580-X). Introduction. Ergodic sets were introd …

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far fr …

The following question came up in a discussion with my advisor: Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-alg …

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable cardinals, …

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real numbe …

Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces. As Gerald Edgar & M …

Sorry for the necromancy. Here's an attempt at constructing a $\sigma$-algebra using the tensor product of $\sigma$-algebras. This should likely not result in a Borel structure (i.e., a $\sigma$-algeb …

Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth? Edit: My original question said "measurable …

One resource you may like is this recent paper by Culbertson and Sturtz on A Categorical Foundation for Bayesian Probability. Here are some thoughts on the category $\mathrm{Meas}$ of measurable sp …