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Let $m$ be Lebesgue measure on $\mathbb R$, and let $m_i$ and $m_o$ be the inner and outer measures respectively. Is it the case that for all $A \subset \mathbb R$ and all $x \in [m_i(A), m_o(A)] …

This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes? Of course when one tries to define measures on "large sets" problems of non …

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard w …

This is a very naive question, maybe more of a reference request than anything else. Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has …

Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated. Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a fil …

This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that …

Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of sub …

Let $\mathcal F$ be a field of subsets of a set $\Omega$. Equip the space $[0,1]^\mathcal F$ of functions from $\mathcal F$ into $[0,1]$ with the product topology. Then, the set $\Delta$ of finitely a …

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't se …

Let $\Omega = \{0,1\}^\mathbb{N}$, let $\mathcal{A}$ be the algebra generated by the open subsets of $\Omega$, where we use the product of discrete topologies, and let $\mathcal{F} = \sigma(\mathcal{A …

Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $ …

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space ($\mathbb P$ is countably additive). Let $\{p_\omega: \omega \in \Omega\}$ be a family of (countably additive) probability measures on $(\O …

The other day I learned of a small error in the book Theory of Charges: A Study of Finitely Additive Measures. Example 11.4.1 goes as follows. Let $\mu_0$ be a finitely additive probability measure de …

I am trying to better understand a condition that appears in Theorem 1 of this paper. Let $K$ be a convex and compact subset of a locally convex tvs. The condition is: $K$ embeds linearly into a stri …

Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\m …