Questions tagged [lebesgue-measure]

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Problem regarding set of positive Lebesgue measure in $\mathbb{R}^2$

Let $T_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set $$L= \bigcup_{p\in[0,1]\cap \mathbb{Q}}T_{p,1} \bigcup_{q\in[0,1]\cap \mathbb{Q}}T_{1,q} $$ and consider $P=[0,1]\times[0,...
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2 votes
1 answer
137 views

$\sigma$-algebra generated by analytic sets

The Borel $\sigma$-algebra $\cal B$ on real numbers has many good properties. For instance, all continuous functions are $\cal B/\cal B$-measurable. On the other side, not only $\cal B$ is not ...
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0 votes
1 answer
117 views

$f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere

Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?
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0 answers
37 views

Sets measurable in every affine subspace

Take a non-measurable subset $S\subseteq [-1,1]$ and subtract $S\times \{0\}$ from the unit disk $B$ in $\mathbb{R}^2$. The set $X=B\setminus (S\times \{0\})$ is measurable by 2-D Lebesgue measure ...
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1 vote
1 answer
132 views

Topological analog of the Lusin-N property

$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets. Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
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0 votes
0 answers
20 views

Properties of probability of a Gaussian random vector being in a linear cone

Let $M$ be a positive non-singular $n \times n$ real matrix. Let $X= (X_1,X_2,\ldots, X_n)$ be a random (column) vector in $\mathbb{R}^n$ with components that are independent and identically ...
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3 votes
2 answers
240 views

Growth of $L^p$ norms as $p \to \infty$

Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
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0 answers
42 views

A heuristic understanding of the theorem for increasing function?

recently I read the following theorem: Suppose $H: R \to R$ is increasing, right continuous, and constant for $x\ge1$ and $x\le0$. Let $\lambda$ be the Lebesgue-Stieltjes measure defined using the ...
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1 vote
1 answer
68 views

Measurable sets of $\mathbb R^n$ forming unique absolutely continuous convex combinations?

If we consider a finite set $A\subset\mathbb R^n$, uniqueness of the convex decomposition of points in $A$ is equivalent to the absence of $\mu\neq0$ signed measure supported on $A$ such that $\mu(\...
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  • 13
1 vote
0 answers
99 views

Do zeroes of $f(t)= \sum_{k\in \mathbb{Z}} e^{\lambda_k t} c_k$, have zero Lebesgue measure ?$\{\lambda_k\}_k$ eigenvalues of elliptic s.a. operator

This question is inspired by the zeroes of solutions to parabolic PDEs (interpret the $\lambda_k$ above as eigenvalues of an elliptic operator), even though I abstracted it from its original context. $...
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0 votes
1 answer
68 views

An equation in the convolution measure algebra on reals

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals. Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
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  • 3,365
4 votes
1 answer
166 views

Generalized limits in Boolean algebras

Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
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0 votes
1 answer
212 views

Is the point-wise limit of simple functions a measurable function?

Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function. Q. Is the point-wise limit of a sequence of simple functions a Borel ...
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1 vote
1 answer
110 views

Isomorphy between Lebesgue space $L_1$ and the $l_1$ sum of $L_1[0,1]$ spaces

Is it true that for an infinite index set $I$, we have that $L_{1}([0,1]^{I}, \mathbb{R})$ can be written as the infinite direct sum of $L_{1}([0,1], \mathbb{R})$, i.e. $$L_{1}([0,1]^{I}, \mathbb{R})=\...
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3 votes
1 answer
229 views

Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not found anything similar to it on the internet. Special version of Tonelli’s theorem Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
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6 votes
1 answer
359 views

A problem concerning a divergent function on $[0, 1]$

This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can'...
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  • 481
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0 answers
32 views

Direction cosine and surface measure

Let G be a bounded finitely connected domain in $\mathbb{R}^m$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature. Each sufficiently small open ...
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  • 159
2 votes
0 answers
96 views

Weak convergence of atomic measure to Lebesgue measure

Let $G$ be the open $n$-ball in $\mathbb{R}^n$ and $G^\Delta$ the set of points in $\mathbb{R}^n$ with distance less than $\Delta>0$ from $G$. Let $G_T=\{Tx: x\in G \}$ and $G_T^\Delta = \{ Tx: x \...
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  • 159
5 votes
0 answers
97 views

Which reals are Lebesgue measures of regions in $\mathbb R^n$ defined by inequalities involving polynomials with integer coefficients?

Let $a$ be a real number. What are necessary and sufficient conditions for the existence of a positive integer $n$ and a finite set of polynomials $p_1,\ldots,p_k$ with integer coefficients in $n$ ...
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1 vote
1 answer
267 views

Support of a measure

Let $T:X\to X$ be a continuous function on a compact manifold $X$ and let $\text{Leb}$ be the Lebesgue measure normalized so that $\text{Leb}(X)= 1$. We denote by $\mathcal{M}(X)$ the space of all ...
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  • 845
0 votes
1 answer
49 views

Evaluating a limit at a discontinuity of a monotone rearrangment (distribution function)

I have a question that occurred to me and has been bothering me, because maybe graphically it seems obvious but I don't know how to get there. It has to do with the distribution function and monotone ...
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  • 113
2 votes
1 answer
92 views

Problem regarding vanishing set of convolution

Let $f$ vanishes on an open set containing 0. So there exists $l>0$ such that $f$ vanishes on $B(0,2l).$ So we can choose $g\in C_c^\infty (\mathbb{R}^n)$ (supported on $B(0,l)$) such that $f*g$ ...
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0 votes
0 answers
26 views

Measure of set of singular points of a generalized discrete Fourier transform

Assume we are given a function that can be expressed in a Fourier-like expansion $$ f\colon \mathcal{X} = [0, 2\pi]^d \to \mathbb{C}, \ \boldsymbol{x}\mapsto \sum_{i = 1}^K c_{i} e^{-i \boldsymbol{\...
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1 vote
0 answers
72 views

Cosine function evaluations are linearly independent? [closed]

Let $\{x_1,\ldots,x_n\}$ be distinct points in $\mathbb{R}^d$, and consider the functions $f_j(x) = cos(w_j^T x + b_j)$ for $w_j \in\mathbb{R}^d$, $b_j\in\mathbb{R}$, $j=1\ldots,m$, and let $m\ge n$. ...
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  • 51
5 votes
1 answer
284 views

Continuity of real functions

The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous? Question Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and ...
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  • 3,771
1 vote
0 answers
83 views

$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]

For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
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  • 11
1 vote
1 answer
71 views

Prove the integral of multi-variable rational fraction is convergent

I have posted this problem in MSE long ago: https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...
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  • 409
1 vote
0 answers
40 views

Pontryagin's principle with Lebesgue-integrable control

Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, ...
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2 votes
2 answers
454 views

In practice, how is the Lebesgue measure usually generalized?

The general question It is easy to find on the Wikipedia page for Lebesgue measure that Haar measure is a common generalization that preserves the idea of "invariance under some group action"...
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  • 451
2 votes
0 answers
72 views

Necessary and sufficient conditions on kernels of trace-class operators

Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$? We know that ...
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  • 143
0 votes
1 answer
67 views

Convoluted Cantor-like measure which has a continuous component [duplicate]

Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable $$ \sum_{k\ge 1}3^{-k}X_k $$...
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  • 1,222
3 votes
1 answer
138 views

Coloring the uncountable Lebesgue-measurable sets of $\mathbb{R}$

A hypergraph $H=(V,E)$ consists of a set $V$ and $E\subseteq {\mathcal P}(V)$, that is, $E$ consists of subsets of $V$ of arbitrary size. Obviously, a graph is a special kind of hypergraph. Let $H=(V,...
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3 votes
1 answer
284 views

Almost surely convergence of translations of a measurable function

Suppose that $\alpha_n$ is a sequence of positive numbers converging to $0$. Question. Is there a bounded measurable function $f$, say $1$-periodic, such that $f_n(x)=f(x-\alpha_n)$ does not ...
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4 votes
1 answer
266 views

Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$

"Measure Theory and Probability Theory" by Athreya and Lahiri introduces Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$ in general in the following way: Let $f:\mathbb{R}^2\...
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  • 143
2 votes
1 answer
393 views

measure of a degenerate Gaussian distribution

I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it in a close form. After starting with a Gaussian random variable and restricting it to a condition, I ...
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4 votes
1 answer
192 views

Vanishing of the product of a function and its own Fourier transform

I have found the following question to be surprisingly hard: Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that $$ f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere}, $$ ...
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  • 1,535
0 votes
0 answers
116 views

Lebesgue measure of a neighbourhood of a curve

Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary). For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
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  • 892
0 votes
1 answer
167 views

Lebesgue measure of sets in $\mathbb{R}^N$

Let $\Omega\subseteq \mathbb{R}^N$ be an open, bounded and connected set (it can be assumed with smooth boundary if necessary). Consider $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (...
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  • 892
0 votes
0 answers
53 views

Sequence of open sets converge in characteristic function to an open set?

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
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  • 892
2 votes
0 answers
314 views

Simple proof of the Lebesgue density theorem in $\Bbb{R}^n$

[I posted this on MSE a while ago, but no answer was forthcoming.] I am looking for a simple proof of the Lebesgue density theorem for $\Bbb{R}^n$. The Wikipedia page on the Lebesgue differentation ...
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1 vote
1 answer
148 views

Is there a maximal translation-invariant extension of Lebesgue measure?

(Cross posted at MSE.) The answer to this question shows that there are translation-invariant extensions of Lebesgue measure. Are there maximal translation-invariant extensions of Lebesgue measure (...
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  • 719
1 vote
1 answer
102 views

Summability issues of measure when we decompose a measurable set into two non-measurable parts

The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\...
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  • 1,449
0 votes
0 answers
129 views

Signed distance function

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function: $d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
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  • 892
1 vote
0 answers
37 views

The Lebesgue measure of the solution space of a specific type of equation

Consider the solution space of parameter $x: D = \{x| F(x) = 0, x \in \mathbb{R}^m\}$, where $F(x) = \sum_{n=1}^{N}a_n \exp\{b_n^T x\}$ for $a_n\in \mathbb{R},b_n \in \mathbb{R}^{m}, n = 1, ..., N$. ...
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1 vote
1 answer
88 views

Existence of integral kernel

I know the following statement ture. Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$. Then, $T$ has the integral kernel $...
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1 vote
1 answer
130 views

Does a subset of positive measure in $\mathbb{R}$ locally "almost" have density $1$? [closed]

Let $A\subseteq \mathbb{R}$ be a Lebesgue-measurable set. We say that $A$ is locally $\varepsilon$-dense if for any $\varepsilon > 0$, there are $x<y\in\mathbb{R}$ such that $$\frac{\mu(A\cap[x,...
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1 vote
0 answers
119 views

The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$

For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$: $$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
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  • 339
15 votes
2 answers
409 views

Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets

Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets. Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
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4 votes
1 answer
880 views

Exchanging series and integrals

I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
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7 votes
3 answers
256 views

Shrinking subset and product

Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
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