# Questions tagged [lebesgue-measure]

The lebesgue-measure tag has no usage guidance.

163
questions

2
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1
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117
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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,...

2
votes

1
answer

137
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### $\sigma$-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 ...

0
votes

1
answer

117
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### $f=0$ 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 $\Omega$?

0
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0
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37
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### 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 ...

1
vote

1
answer

132
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### 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 $\...

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 ...

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 \...

0
votes

0
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42
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### 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 ...

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(\...

1
vote

0
answers

99
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### 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.
$...

0
votes

1
answer

68
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### 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 ...

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 ...

0
votes

1
answer

212
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### 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 ...

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})=\...

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{...

6
votes

1
answer

359
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### 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'...

0
votes

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 ...

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 \...

5
votes

0
answers

97
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### 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 $n$ ...

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 ...

0
votes

1
answer

49
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### 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 ...

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 $f*g$ ...

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{\...

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 $m\ge n$. ...

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 ...

1
vote

0
answers

83
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### $ \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)}...

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 ...

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, ...

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"...

2
votes

0
answers

72
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### 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 ...

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
$$
\sum_{k\ge 1}3^{-k}X_k
$$...

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,...

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 ...

4
votes

1
answer

266
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### 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\...

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 ...

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
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
...

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}...

0
votes

1
answer

167
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### 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 $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (...

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 ...

2
votes

0
answers

314
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### 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 ...

1
vote

1
answer

148
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### 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 (...

1
vote

1
answer

102
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### 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) +\...

0
votes

0
answers

129
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### 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,\...

1
vote

0
answers

37
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### 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 $a_n\in \mathbb{R},b_n \in \mathbb{R}^{m}, n = 1, ..., N$.
...

1
vote

1
answer

88
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### 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 $...

1
vote

1
answer

130
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### 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,...

1
vote

0
answers

119
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### 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$...

15
votes

2
answers

409
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### 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}...

4
votes

1
answer

880
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### 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 ...

7
votes

3
answers

256
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### 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 $|...