# Questions tagged [measure-theory]

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

1,697 questions

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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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### 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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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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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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### 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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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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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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### 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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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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### 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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### 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 $\mu_j$ ...