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Questions tagged [lebesgue-measure]

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0
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48 views

convergence of a fraction containing Lebesgue measures and perimeters

Sorry for this basic question posted on MO, I as well asked here https://math.stackexchange.com/questions/3063202/convergence-of-a-fraction-containing-lebesgue-measures-and-perimeters Let $\epsilon&...
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2answers
73 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 ...
2
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0answers
86 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\...
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3answers
405 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\...
2
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1answer
160 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\...
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2answers
359 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 ...
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0answers
116 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 ...
2
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1answer
100 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 ...
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1answer
135 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 ...
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1answer
83 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 ...
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1answer
81 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 ...
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5answers
1k 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 ...
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6answers
1k 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 $\...
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0answers
246 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?
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1answer
162 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{...
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1answer
89 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 \...
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1answer
78 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 ...
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0answers
47 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 \...
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1answer
81 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 ...
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2answers
212 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 ...
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0answers
135 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 $...
3
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1answer
127 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 $B_\epsilon(a)$ ...
4
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0answers
153 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 ...
7
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2answers
226 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 ...
13
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1answer
418 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 ...
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2answers
187 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 $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
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0answers
137 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 ...
2
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1answer
124 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:\...
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2answers
359 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 $...
2
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1answer
305 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 ...
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2answers
1k 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 ...
8
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1answer
96 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 ...
3
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0answers
147 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 $A\subset \mathbb{R}^n$ ...
10
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1answer
273 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 $i \in I$...
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1answer
266 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....
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1answer
74 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 ...
3
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1answer
307 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 ...
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2answers
252 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_{\...
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3answers
2k 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 ...
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2answers
636 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 $w'(p)$ ...
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2answers
127 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,...
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2answers
445 views

Can a nowhere differentiable function preserve measurability?

I want to know whether a continuous nowhere differentiable function $f: \mathbf{R} \to \mathbf{R}$ can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know ...
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8answers
5k views

Physical meaning of the Lebesgue measure

Question (informal) Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the ...
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1answer
95 views

Set of General Linear Position with Nonzero Measure

I came to the following question, but I don't have quite a good idea how to approach. Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position? I believe that,...
4
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1answer
286 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
2
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1answer
100 views

Lebesgue measurability of singular set

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function. Define a superdifferential of $f$ at $x\in Q$ by $$ D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid \text{$f(y)\...
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vote
1answer
138 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
0
votes
1answer
124 views

Approximating characteristic functions by cutting the real axis into smaller and smaller pieces

Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
6
votes
0answers
145 views

orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
8
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2answers
884 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?