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It is well known that a finite measure on the Baire $\sigma$-algebra of a, say, compact Hausdorff space can be extended to a unique regular measure defined on the Borel $\sigma$-algebra. The Baire $\s …

I'm trying for some time now to prove or disprove the following conjecture to no avail: Let $S$ be a set and let $(\Sigma _n)$ be a sequence of countably generated $\sigma$-algebras on $S$ sat …

Let $(\Omega,\Sigma,\mu)$ be a probability space. A $\mu$-atom is an $A\in\Sigma$ such that $\mu(A)>0$ and for all $B\in\Sigma$ such that $B\subseteq A$, either $\mu(B)=\mu(A)$ or $\mu(B)=0$ holds. N …

Is it consistent that there exists a nonzero atomless finite measure on some $\sigma$-algebra on a cardinal $\kappa$ satisfying $\kappa<\mathfrak{c}$? Can there be such a measure on $\omega_1$ …

Let $X$ and $Y$ be Polish spaces, let $\mathcal{P}(Y)$ be the space of Borel probability measures on $Y$ endowed with the smallest $\sigma$-algebra such that all functions of the form $\nu\mapsto\nu(A …

Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability mea …

Let $(\Omega,\Sigma,\mu)$ be a probability space, $(X,\mathcal{X})$ be a measurable space and $R(\Omega,X)$ be the set of equivalence classes of measurable functions from $\Omega$ to $X$ under almost …

Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ contai …

Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all …

Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\subset …

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a se …

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact metrizable space endowed with its Borel $\sigma$-algebra $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally measu …

The following question has puzzled me for some time: Let $(\Omega,\Sigma)$ be a nonempty, measurable space. Does there necessarily exist a probability measure $\mu:\Sigma\to[0,1]$? If the …

Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \ …