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        Questions tagged [measure-theory]

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

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        Relationship of Baire sets and Baire functions

        Halmos Measure Theory has a problem (51.6) which goes as follows: The term "Baire set" is suggested by the term "Baire function" as used in analysis. If $\mathscr{B}$ is the smallest class of ...
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        155 views
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        Polish transversals

        A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$. So a continuum has a composant transversal precisely when ...
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        Reference Request: Korevaar and Schoen Spaces

        Let $(X,d,m)$ and $(Y,\rho,\mu)$ be doubling metric measure spaces. The seminal work of Korevaar and Schoen discussed generalizations of $L^p$ spaces for maps from $X$ to $Y$. Standard results from ...
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        283 views

        Partitioning $\mathbb{R}^n$ into closed sets

        Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected. Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
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        215 views

        Concerning Luzin-(N)-property

        Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
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        3answers
        370 views

        Arithmetically random bitstreams

        Motivation (informal). When trying to generate a random bit-stream, we expect that "half of the" bits are $0$, and the "other half" are $1$. So, how about $010101\ldots$? Well, we would also expect ...
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        1answer
        280 views

        Why is it difficult to solve the Monge problem directly?

        I'm trying to understand something about the Monge problem. The Monge problem is: Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
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        1answer
        100 views

        A question about measure-weighted barycenters

        This is a question taken (inferred) from Ex. 19, chap 3 in Rudin's Real and Complex Analysis book. Let $\mu$ be Legesgue's measure on $X=[0,1]$. Given a measurable $L^{\infty}$ function $f:X\to C$, ...
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        1answer
        81 views

        Control the oscillation of a function by its total variation

        Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?
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        Points of continuity of Kullback-Leibler divergence with respect to weak convergence

        I know that the Kullback-Leibler $D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$ over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
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        1answer
        237 views

        Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

        I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
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        31 views

        Mean ergodicity for lifted vector field

        Let $X \in C^{\infty}(TM)$ be an ergodic vector field on a smooth compact manifold and $f$, $\mu$ a function/measure on $M$ satisfying $\mathcal{L}_X \mu =f\mu$. Consider the lifted vector field $$\...
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        1answer
        76 views

        Relation between the measures of two sets defined via Lebesgue integration

        I posted this question on StackExchange, people have upvoted it but I have not received any response. I read up here that it is okay to post unanswered StackExchange questions on Mathoverflow. So, ...
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        52 views

        Energy-minimizing set of discrete points in a bounded domain

        Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all ...
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        117 views

        Measure theory problem concerning convergence of integrals

        Let $X$ be a measure space. Let $S_j$, $j \in \mathbb N$ be an increasing sequence of $\sigma$-algebras on $X$ such that $S := \bigcup_{j \geq 0} S_j$ is a $\sigma$-algebra. For every $j$, let $\mu_j$ ...

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