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Here's a concrete example of an atomless measure. Let $f \in L^1$ be an integrable function with total mass 1 (i.e. $\int_0^1 f = 1$). Define $$\mathbb P(A) = \int_A f(x) ~dx$$ for any Borel set $A$ …

Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ i …

Here's an answer in the case that $X$ and $Y$ are Gaussians. There's such a rigid algebraic structure there that I don't know whether I'm exploiting that, or a more general property of the distributi …

Let's assume that we are working with the canonical probability space $\Omega = D(\mathbb R)$ of càdlàg functions, and $\mathbb P$ is the law of the process. I would doubt that there is a satisfactor …

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 …

Since a troll bumped this question to the front page, I might as well answer it. The technology which provides the solution is called regular conditional probability or disintegration.

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 …

This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?. Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space …

Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure descri …

Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the mea …

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 …

Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ …

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of p …

Here is a candidate class of examples. I have made this community wiki so please feel free to edit it if you can answer it. Or, copy the text and make a new answer so we can give you reputation points …

It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality. Hard question: What are the most general structures on which one may define something whi …