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I know in computable analysis, which is closely related to the descriptive set theoretic questions you are asking, that the Lévy–Prokhorov metric on the weak topology is useful. (I think the narrow t …

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is invaria …

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a me …

Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ a …

If you clarify your question, I can modify this answer to be better. (I am sure you won't like my answer. The best answer is this is not practical, except in the simplest of settings. Or you are in …

I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write …

Short question If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked …

If $X$ and $Y$ are Polish and $Y$ is compact, then YES. (I think my proof can be fixed to handle the noncompact setting, but I don't see how right now.) My proof involves this lemma. Lemma. Assume …

I can't explain the group theoretic structure of $\Gamma$, but I can explain the topological structure. (Warning, this post is mostly a continuous stream of thoughts. I hope it is well organized and …

The short answer is it depends on what one means by extending Kolmogorov complexity. The details are below. First, when I say Kolmogorov complexity, I will mean prefix-free complexity since that is …

(My understanding of this material has significantly gone up in the months since I asked it, and I will attempt to answer my own question.) In general, if $(\Omega,\mathcal{B},\mathbb{P},\{T_g\})$ is …

Background I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since …

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \m …

For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which co …