# Questions tagged [lebesgue-measure]

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### Regarding a positive Lebesgue measure set in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure. ...
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### Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
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### Measure of the boundary of an BV-exention domain: Do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
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### Sharp assumption for preserving Lebesgue measurability by left composition

Let $g: [0, 1] \to \mathbb R$ be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that $f \circ g$ ...
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### Problem regarding Lebesgue measure in $\mathbb{R}^2$

Let $P=A_1\times A_2,$ where $A_1,A_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue ...
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### Computing the infinite dimensional Lebesgue measure of “cubes”

There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
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### Continuity of the Lebesgue measure w.r.t the Hausdorff metric

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff ...
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### Coincidence Topologies for $L^p$ spaces

If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
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### Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
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### Volume computation using probabilistic approach

Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i....
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### Functions that are almost (left-) continuous almost everywhere

Denote the Lebesgue measure on $[0, T]$ as $\lambda(\cdot)$. Call a measurable function $f : [0, T] \to \mathbb{R}$ almost left-continuous almost everywhere if there exists an $A \subseteq [0, T]$ ...
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### On tensor products of (essentially) bounded measurable functions

Let $F:\mathbb R^2\rightarrow\mathbb R$ be an essentially bounded measurable function ($\mathbb R^d$ is equipped with its standard Lebesgue measure) and assume that $F(x,y)=F(y,x)$. I would like to ...
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### On the existence of a family of countably additive extensions of Lebesgue measure

Let $m$ be Lebesgue measure on $\mathbb R$, and let $m_i$ and $m_o$ be the inner and outer measures respectively. Is it the case that for all $A \subset \mathbb R$ and all $x \in [m_i(A), m_o(A)]$ ...
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### Estimate Integral w.r.t Dirac measure by Integral w.r.t Lebesgue measure

Given a bounded, nonnegative and measurable function $f$ and denote the Euclidean ball with center $z$ and radius $r$ by $B_r(z)$. Is it somehow possible to estimate the Integral w.r.t. the Dirac ...
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### Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem. Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...
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### A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?

Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power. Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...
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### How to rigorously define a translation-invariant measure that follows these requirements?

I am unsure anyone at math stack exchange can answer my question, so I moved it to MathOverflow. Let $A$ be a subset of $\mathbb{R}$. I want to rigorously define what I believe is the most "intuitive ...
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### Integrability of log of distance function

Let $E\subset B_1(0)\subset \mathbb{R}^n$ be a compact set s.t. $\lambda(E)=0$, where $\lambda$ is the Lebesgue measure, and $B_1(0)$ is the Euclidean unit ball centered at the origin. Is the ...
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### Interpolation Inequality's Proof

Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$: \begin{equation} \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
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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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### A measurable set such that its intersection and difference with every interval have the same measure [duplicate]

Let $\Omega = [0,1]$. I want a Lebesgue measurable set $S$ with the following property. $$\ell(S \cap I) = \ell(I \backslash S)$$ for every subinterval $I$ of $[0,1]$, where $\ell(A)$ is the ...
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### Elliptic equation with lower dimensional data

I'm looking at $u - \Delta^2 u = f$ with homogeneous boundary and Neumann conditions on the unit square, $\Omega$. In particular, I'm looking at the case where $f\in L^2(S)$ is only supported on a ...
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### Injection of Besov spaces in $L^p$

I believe that for $p\ge 2$, we have the continuous injection (for $p=2$, it is an equality), $$B^0_{p,2}(\mathbb R^n)\subset L^p(\mathbb R^n),$$ where $B^0_{p,2}(\mathbb R^n)$ is the Besov space. ...
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### Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
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### Positive part of “outer sums” of measures

Here is a question about decomposition of measures in singular parts and in positive and negative parts. $\newcommand{\RR}{\mathbb{R}}$ Let $\Omega_{1/2}$ be compact subsets of $\RR^d$ equipped ...
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### Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
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### automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...
### Surface/Volume-Ratio of an $\epsilon$-extension of a compact subset $S \subset \mathbb R^n$
For a non-empty, compact set $S \subset \mathbb{R}^n$, the $\epsilon$-extension of $S$, $S_\epsilon$, is defined to be the set $$S_\epsilon = \cup_{a \in A} B_{\epsilon}(a),$$ where $B_\epsilon(a)$ ...