Questions tagged [lebesgue-measure]

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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 ...
Fabrice Pautot's user avatar
6 votes
2 answers
379 views

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 ...
JustSomeGuy's user avatar
-1 votes
1 answer
368 views

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\...
Elena Villalón Sepúlveda's user avatar
1 vote
1 answer
112 views

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, ...
avk255's user avatar
  • 543
0 votes
1 answer
853 views

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 ...
avk255's user avatar
  • 543
1 vote
1 answer
55 views

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 ...
turtlesandwich's user avatar
1 vote
1 answer
421 views

Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere

Let $f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$ $g:=\ln f$ (and assume $g'$ is Lipschitz continuous) $n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
0xbadf00d's user avatar
  • 161
3 votes
0 answers
147 views

Does there exist a compactly supported integrable function with infinite Coulomb energy?

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that $$ E[f] = \iint\limits_{\Omega\...
Ben Ciotti's user avatar
3 votes
1 answer
442 views

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. ...
Bazin's user avatar
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4 votes
1 answer
374 views

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 $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
pitariver's user avatar
  • 267
1 vote
2 answers
112 views

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 ...
Dirk's user avatar
  • 12.3k
2 votes
0 answers
246 views

Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
Keshav Srinivasan's user avatar
8 votes
3 answers
1k views

For what sets does the Lebesgue Differentiation Theorem hold in one dimension?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
Keshav Srinivasan's user avatar
2 votes
1 answer
894 views

Does the Lebesgue Differentiation Theorem hold for regular polytopes?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
Keshav Srinivasan's user avatar
16 votes
2 answers
1k views

Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?

Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...
Keshav Srinivasan's user avatar
0 votes
0 answers
528 views

Egorov's and Lusin's Theorem in the space with infinite measure

Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite. On the measurable space whose measure is infinite, does there ...
ABB's user avatar
  • 3,898
2 votes
1 answer
115 views

Measure preserving coordinates of $S^2$ from $[0,1]^2$

Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively. Question ...
Amir Sagiv's user avatar
  • 3,544
-2 votes
1 answer
1k views

Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
Saj_Eda's user avatar
  • 395
1 vote
1 answer
240 views

Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...
Saj_Eda's user avatar
  • 395
3 votes
1 answer
90 views

Measurability of specific function

Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval. Since ...
sgr's user avatar
  • 65
21 votes
6 answers
4k views

Why is Lebesgue measure theory asymmetric?

A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer ...
user avatar
21 votes
6 answers
4k views

Lebesgue measure theory applications

I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example. Theorem: Let $X$ be a differentiable submanifold of $\...
Eduardo's user avatar
  • 757
2 votes
0 answers
396 views

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?
Zahra's user avatar
  • 29
3 votes
1 answer
406 views

Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?

Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$. Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows: $$\mathcal{...
user173856's user avatar
  • 1,987
1 vote
1 answer
142 views

Convergence of measurable functions in a locally compact space

Set $(X,\mathcal{B})$ a measurable space. If $f:X\rightarrow[0,\infty)$ is a measurable function then exists a sequence of simple functions $\{s_n\}_{n\geq1}$ such that $$0\leq s_1 \leq s_2\leq \...
Rusbert's user avatar
  • 173
4 votes
1 answer
168 views

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 ...
Exodd's user avatar
  • 201
2 votes
0 answers
772 views

Existence of unbounded $M \subset \Bbb{R}$ of finite measure s.t. $1_M$ is $L^p$-Fourier multiplier

I would like to know if there is a measurable set $M \subset \Bbb{R}$ such that $M$ has finite Lebesgue measure $0 < \lambda(M) < \infty$, $M$ is unbounded in the sense that $\lambda(M \...
PhoemueX's user avatar
  • 754
0 votes
1 answer
133 views

Change of variables for double integral [closed]

Thank you for your time. My basic question is whether the following change of variables allowed $$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$ I fail to ...
Xing Wang's user avatar
7 votes
2 answers
851 views

If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.?

If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that ...
ohliv's user avatar
  • 73
0 votes
0 answers
286 views

When convolution with exponential kernel is bounded

Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying $...
Saj_Eda's user avatar
  • 395
4 votes
1 answer
258 views

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 $B_\epsilon(a)$ ...
Lucas L. 's user avatar
5 votes
0 answers
206 views

Existence of $A\subset\Bbb{R}^n$ of finite measure and $\hat{1_A}\in\bigcap_{q>1}L^q$, but s.t. for some $1<p<\infty$, $1_A$ is no $L^p$-Fourier mult

I am interested in the following somewhat obscure question: Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its ...
PhoemueX's user avatar
  • 754
7 votes
2 answers
255 views

$f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$

Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim _{r\to \infty} \dfrac 1r \int_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for ...
user avatar
24 votes
3 answers
1k views

Average measure of intersection of a convex region with its translate

Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region. My question is about $$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ How ...
zref's user avatar
  • 343
4 votes
2 answers
225 views

lower bound volume of a set

Let $\lambda$ be Lebesgue measure on [0,1]. For any $x_{1},\dots,x_{k}$ in $[0,1]$, define $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
Cuize Han's user avatar
3 votes
0 answers
169 views

Random sets and invariant extensions of Lebesgue measure

Given AC, is there a probability measure $\mu$ on $2^{[0,1]}$ and a translation-invariant extension $\lambda$ of Lebesgue measure on $[0,1]$ such that: for all permutations $\pi$ of $[0,1]$ and all ...
Alexander Pruss's user avatar
2 votes
1 answer
229 views

Measurability of a parametrized conditional expectation

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ a Sub-$\sigma$-Algebra. Moreover, let $X:\Omega\rightarrow\mathbb{R}$ be a random variable and $F:\...
Mark's user avatar
  • 169
6 votes
2 answers
2k views

Extending the Lebesgue measure

The Lebesgue measure $\lambda$ is a function on a subset of the power set of real numbers $\mathbb{R}$ that satisfies the following properties (among others): (i) $\lambda$ is finitely additive: If $...
D. Ror.'s user avatar
  • 399
3 votes
1 answer
1k views

Whether $\varphi(E)$ is a Jordan measurable set?

Definition: A set $S \subset \mathbb {R^{n}}$ is Jordan measurable if it is bounded in $\mathbb {R^{n}}$ and its boundary is a set of Lebesgue measure zero. The following conclusion has been ...
james  givens's user avatar
13 votes
2 answers
3k views

Radon-Nikodym theorem for non-sigma finite measures

Let $(X,\mathcal M, \mu)$ be a measured space where $\mu$ is a positive measure. Let $\lambda$ be a complex measure on $(X,\mathcal M)$. When $\mu$ is sigma-finite, the Radon-Nikodym theorem provides ...
Bazin's user avatar
  • 15.1k
8 votes
1 answer
106 views

Infering shapes from overlap with a shifting circle

A recent episode of Star Talk Radio discussed among other things the unknown object(s) orbiting Tabby's star (aka "Alien mega structure discovered!" in non-scientific media) and an astronomer said ...
Johannes Hahn's user avatar
4 votes
1 answer
719 views

Lebesgue-Besicovitch theorem for partition elements rather than balls

I'll state the classic result in its density (rather than the more general differentiation) version. Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ and $A\subset \mathbb{R}^n$ ...
Aryeh Kontorovich's user avatar
11 votes
1 answer
322 views

Partition into sets of positive outer measure

Let $\mu^{\star}$ denote Lebesgue outer measure. Suppose $X \subseteq [0, 1]$ and $\mu^{\star}(X) > 0$. Can we divide $X$ into uncountably many sets $\{X_i : i \in I\}$ such that for every $i \in I$...
Lebesgue's user avatar
  • 131
14 votes
1 answer
529 views

Does the existence of a non-principal measure on ω imply that of a non Lebesgue measurable set?

A non-principal [probability] measure on a set X is a function $\mu$ defined on all subsets of $X$, with values in $[0,1]$, which is finitely additive, satisfies $\mu(X)=1$, and vanishes on singletons....
Jean-Claude Sikorav's user avatar
4 votes
1 answer
116 views

Integral form of maximal function estimate on variable exponent spaces

I am trying to show an estimate of the following form: Given any $p(x)$ such that $1<p^-\leq p(x) \leq p^+ <\infty$ and $p(\cdot)$ is log-Holder continuous, does there exists an $R_0$ (depending ...
Adi's user avatar
  • 483
4 votes
1 answer
987 views

Product of two non-measurable sets

Let $A\subset\mathbb{R}^p$ and $B\subset\mathbb{R}^q$, it’s not difficult to show that $$m^*(A\times B)\leq m^*(A)\cdot m^*(B)$$, where $m^*()$stands for the outter measure in Lebesgue meaning. If A ...
Yijun Yuan's user avatar
3 votes
2 answers
443 views

Differentiate a growing volume

Let me motivate my question with this example. The volume integral of a ball $\int_{B(0,R)} dx$ can be written as an integral over the surface of balls, i.e. $$\int_{B(0,R)} dx = \int_0^R \int_{\...
Frederique's user avatar
19 votes
3 answers
3k views

Measure induced on [0, 1] by infinite tosses of biased coin

It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary. I was ...
Anindya's user avatar
  • 665
2 votes
2 answers
1k views

Differentiate an integral (Lebesgue integral)

Let $f:[0,1]\to\mathbb{R}$ be a bounded (Lebesgue) measurable function. Consider the function $$w(p)=\int_0^1|f|^p\,d\mu$$. Is $w(p)$ differentiable at any $0<p<\infty$? I.e. does $w'(p)$ ...
yoyostein's user avatar
  • 1,219
2 votes
2 answers
224 views

Is the domain of symmetric derivative borel set?

Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$. Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
Rubertos's user avatar
  • 337