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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.

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

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