Questions tagged [lebesgue-measure]

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Analyticity of a function in two complex variables

Let $f$ be a function defined on $\mathbb{C}^2$ given by $$ f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
Aniruddha 's user avatar
3 votes
1 answer
133 views

Recover an $L^1$ integrand by partial differentiation

Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by $$F(x,y) = ...
w116c576's user avatar
1 vote
1 answer
109 views

Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
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1 answer
204 views

What is the measure of two sets which partition the reals into subsets of positive measure?

This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$. (In ...
Arbuja's user avatar
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5 votes
1 answer
319 views

Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?

Suppose $\lambda^{*}$ is the Lebesgue outer measure. Question: Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where: The range of $f$ is $\mathbb{R}$ For all real $x_1,x_2,y_1,y_2$, where $...
Arbuja's user avatar
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2 votes
1 answer
134 views

Finding an explicit & bijective function that satisfies the following properties?

Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$. Question: Does there exist an explicit ...
Arbuja's user avatar
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Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support

In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
CoffeeArabica's user avatar
0 votes
1 answer
75 views

Convergence in sequential Lebesgue spaces

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
Guy Fsone's user avatar
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The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence

Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
David Gao's user avatar
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12 votes
3 answers
622 views

If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...
David Gao's user avatar
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8 votes
0 answers
194 views

*-homomorphisms from $L^\infty(0, 1)$ to itself that acts as the identity on continuous functions

Let $\pi: L^\infty([0, 1], \lambda) \rightarrow L^\infty([0, 1], \lambda)$ be a *-homomorphism (where $\lambda$ is the Lebesgue measure on $[0, 1]$), not necessarily normal (otherwise the question is ...
David Gao's user avatar
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13 votes
1 answer
568 views

Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH

Assuming the negation of CH, let $\omega_1$ be the first uncountable ordinal, $\mathfrak{c}$ be the cardinality of the continuum. Does there exist a map $f: \omega_1 \times [0, 1] \rightarrow \...
David Gao's user avatar
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4 votes
0 answers
142 views

Naïve definition of a measure on a fractal

This question was previously posted on MSE. Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$. One option would be to use ...
Matheus Manzatto's user avatar
1 vote
1 answer
119 views

Quantitative version of Lebesgue points theorem

Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
tommy1996q's user avatar
1 vote
0 answers
35 views

The limit set of consecutive applications of linear transforms to the single segment

Problem. Consider $n$ positive integers $1 < a_1\le \ldots \le a_n$ and $I = \left[\frac{1}{a_n - 1}, \frac{1}{a_1 - 1}\right]$. For each $a_k$ define the linear transform $\phi_k\colon x\mapsto \...
Pavel Gubkin's user avatar
1 vote
0 answers
42 views

Measurability in a product space of a set defined only along its fibers

Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
Giuseppe Tenaglia's user avatar
6 votes
1 answer
173 views

Concentration of volume towards the boundary

Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let $$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$ be the set of all ...
nullptr's user avatar
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4 votes
0 answers
227 views

If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant? Context: I was ...
Saúl RM's user avatar
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26 votes
2 answers
2k views

What is the origin/history of the following very short definition of the Lebesgue integral?

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
Gro-Tsen's user avatar
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3 votes
1 answer
161 views

Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
user483450's user avatar
5 votes
0 answers
112 views

Is there a natural finitely additive measure for which Vitali sets have measure zero?

Vitali sets are nonmeasurable and in particular are not null sets. But all Vitali sets are in some sense small, as described below. Let $V$ be any Vitali set and let $k \in \mathbb{N}$. For each $i \...
Aaron Hill's user avatar
6 votes
0 answers
254 views

Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
Paz's user avatar
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8 votes
1 answer
2k views

How badly can the Lebesgue differentiation theorem fail?

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that $$ \lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ? $$ This is obvious if $0$ is a Lebesgue point ...
No-one's user avatar
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2 votes
1 answer
156 views

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,...
user483450's user avatar
2 votes
1 answer
238 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 ...
Giafazio's user avatar
  • 163
0 votes
1 answer
135 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 $\Omega$?
Billal Elhamza's user avatar
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0 answers
43 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 ...
Brendan McKay's user avatar
1 vote
1 answer
156 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 $\...
Sourav Ghosh's user avatar
3 votes
2 answers
389 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 \...
Hammerhead's user avatar
  • 1,141
1 vote
1 answer
95 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(\...
Cryme's user avatar
  • 13
0 votes
1 answer
79 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 ...
ABB's user avatar
  • 3,794
4 votes
1 answer
184 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 ...
Damian Sobota's user avatar
0 votes
1 answer
383 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 ...
ABB's user avatar
  • 3,794
1 vote
1 answer
125 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})=\...
user44155's user avatar
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3 votes
1 answer
277 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{...
Mr. Proof's user avatar
  • 141
6 votes
1 answer
377 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'...
Ivan's user avatar
  • 491
2 votes
0 answers
121 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 \...
HyyFly's user avatar
  • 187
5 votes
0 answers
98 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 $n$ ...
Mike Krebs's user avatar
1 vote
1 answer
355 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 ...
Adam's user avatar
  • 970
0 votes
1 answer
59 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 ...
didiegop's user avatar
  • 113
2 votes
1 answer
97 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 $f*g$ ...
Wilderness's user avatar
1 vote
0 answers
76 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 $m\ge n$. ...
KNW's user avatar
  • 51
5 votes
1 answer
288 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 ...
喻 良's user avatar
  • 4,076
1 vote
0 answers
88 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)}...
YC Z's user avatar
  • 11
1 vote
1 answer
94 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 ...
Houa's user avatar
  • 519
1 vote
0 answers
50 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, ...
David Ketcheson's user avatar
2 votes
2 answers
582 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"...
exfret's user avatar
  • 479
2 votes
0 answers
116 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 ...
Nemis L.'s user avatar
  • 143
0 votes
1 answer
103 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 $$ \sum_{k\ge 1}3^{-k}X_k $$...
kaleidoscop's user avatar
  • 1,268
3 votes
1 answer
150 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,...
Dominic van der Zypen's user avatar