# Questions tagged [measure-theory]

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

375 questions
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### A subset of plane that meets every line at length one

Is there a subset of plane whose intersection with every line has length one? It is easy to construct such a set under the continuum hypothesis. Also, no such set is Lebesgue measurable - See ...
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### Lebesgue density 1/2 (or bounded away from 0 and 1)

From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
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### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
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### Estimation of the Gromov–Wasserstein distance of spheres

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...
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### Covering inequality for sets of intervals

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point ...
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### Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
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### Quotients of Measurable Spaces?

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 $\Pi$ ...
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### Weak$^*$ convergence of measures vs. convergence of supports

Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
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### A Banach-Tarski game

This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written. A paradoxical family of subsets ...
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### Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...
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### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space: Let $X_1,X_2,\cdots$, be ...
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### A small planar set containing a large family of curves

A beautiful construction by Besicovitch and Rado  produces an astounding example of a compact connected plane set of measure zero containing circles of all radii $r\in(0,1]$. A corollary to a ...
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### Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability ...
It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...