# Tagged Questions

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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### Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
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### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
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### Sigma algebra generated by SOT versus of sigma algebra generated by WOT

Let $H$ be a non-separable Hilbert space. Let us denote $B_s$ ($B_w$), by the sigma algebra generated by the strong operator topology (weak operator topology) on $B(H)$. Question: Is $B_s$ the same ...
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### When is Hausdorff measure a Frostman measure?

Let $(X,d)$ be a metric space and let $\mathcal{H}^s$ be the $s$-dimensional Hausdorff measure on $X$. For a measure $\mu$ on $X$, we say that $\mu$ is a Frostman measure (sometimes referred as ...
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### Operator-valued measurable functions

Let $H$ be a non separable Hilbert space and $\Omega$ be a measurable space. Naturally, we say that $f:\Omega\to B(H)$ is $w$-measurable if $f^{-1}(O)$ is measurable for any open set $O$ in the weak ...
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### Number of lattice points in homotetic image

I asked this question on MSE a week ago and it gave me a tumbleweed badge :-) Let $\Lambda$ be a lattice in $\mathbb R^n$, with covolume $\Gamma$. Moreover, let $S$ be a bounded (Lebesgue-)measurable ...
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### Moment Sequence in l²

I have the following problem/question: For which finite regular complex measures $\mu$ is the moment sequence $$\left(\int_{[-1,1]}t^k\,d\mu\right)_{k\in\mathbb N}$$ a member of $\ell^2(\mathbb N)$?...
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### What are universal abstract $\sigma$-algebras on $\sigma$-frames?

Originally asked on MSE. In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice ...
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### On the use of the term “field of sets” in Maharam's papers

I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for ...
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### Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?

A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on ...
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### Stochastic equation

Let $X,Y$ be Polish spaces and $\kappa:X\times \mathcal B(Y)\to[0,1]$ be a Borel-measurable stochastic kernel on $Y$ given $X$. Under which conditions for a probability measure $\nu$ on $Y$ there ...
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### A boundary of the second fundamental theorem of calculus

Let's say that a set $X\subseteq [0,1]$ has Property Q if the following holds: For every continuous $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ and derivative existing and bounded by 1 on $[0,1]\setminus X$, ...
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### How to prove that a monotone function is differentiable at some point?

This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). Is there a significantly different approach? Let me ...
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### List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)? I am aware that it is known for some uniformly ...
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