Questions tagged [lebesgue-measure]

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25 views

Dominance convergence theorem to compute expectation of a sequence of random variables defined by their time derivatives

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Then consider a sequence $X_t^0,X_t^1,\ldots, X_t^n$ for which we get $Y_t^0,Y_t^1,\ldots, ...
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0answers
62 views

Regarding normalized Lebesgue measure

Let $\Omega$ be a compact Hausdorff space. Let $ \partial \mathbb{D}$ be the boundary of the unit circle in the complex plane $\mathbb{C}$. Let $\mu$ be a positive Radon Measure on $\Omega$ with $\mu(\...
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0answers
63 views

Coincidence Topologies for $L^p$ spaces

If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
11
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1answer
149 views

Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
2
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1answer
123 views

Volume computation using probabilistic approach

Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i....
2
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0answers
44 views

Functions that are almost (left-) continuous almost everywhere

Denote the Lebesgue measure on $[0, T]$ as $\lambda(\cdot)$. Call a measurable function $f : [0, T] \to \mathbb{R}$ almost left-continuous almost everywhere if there exists an $A \subseteq [0, T]$ ...
0
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1answer
56 views

On tensor products of (essentially) bounded measurable functions

Let $F:\mathbb R^2\rightarrow\mathbb R$ be an essentially bounded measurable function ($\mathbb R^d$ is equipped with its standard Lebesgue measure) and assume that $F(x,y)=F(y,x)$. I would like to ...
14
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1answer
425 views

On the existence of a family of countably additive extensions of Lebesgue measure

Let $m$ be Lebesgue measure on $\mathbb R$, and let $m_i$ and $m_o$ be the inner and outer measures respectively. Is it the case that for all $A \subset \mathbb R$ and all $x \in [m_i(A), m_o(A)]$ ...
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0answers
61 views

Estimate Integral w.r.t Dirac measure by Integral w.r.t Lebesgue measure

Given a bounded, nonnegative and measurable function $f$ and denote the Euclidean ball with center $z$ and radius $r$ by $B_r(z)$. Is it somehow possible to estimate the Integral w.r.t. the Dirac ...
14
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1answer
497 views

Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem. Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...
43
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4answers
5k views

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 ...
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1answer
658 views

How to rigorously define a translation-invariant measure that follows these requirements?

I am unsure anyone at math stack exchange can answer my question, so I moved it to MathOverflow. Let $A$ be a subset of $\mathbb{R}$. I want to rigorously define what I believe is the most "intuitive ...
6
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2answers
268 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 ...
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0answers
38 views

Quantify the decay of push-forward measures of points with small values

Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, and $u:\Omega\to \mathbb{R}$ is a Lebesgue measurable function. Then for each $\epsilon>0$, we define the set $$ A_\epsilon=\{x\in \Omega\...
-1
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1answer
129 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\...
1
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1answer
88 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, ...
0
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1answer
153 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 ...
1
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1answer
49 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 ...
1
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1answer
345 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(...
3
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0answers
110 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\...
3
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1answer
156 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. ...
3
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1answer
200 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 $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
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2answers
90 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
115 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
536 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
323 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\...
15
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2answers
697 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
303 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
106 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
422 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 ...
1
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1answer
146 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 ...
3
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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 ...
18
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5answers
2k 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
3k 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 $\...
1
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0answers
254 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?
3
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1answer
234 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{...
1
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1answer
101 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 \...
4
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1answer
97 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 ...
2
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0answers
141 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 \...
0
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1answer
95 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 ...
7
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2answers
297 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 ...
0
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0answers
192 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 $...
4
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1answer
192 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 $B_\epsilon(a)$ ...
5
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0answers
163 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
votes
2answers
236 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 ...
21
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2answers
830 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 ...
4
votes
2answers
204 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 $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
3
votes
0answers
146 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
votes
1answer
152 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:\...
3
votes
2answers
667 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 $...