Questions tagged [lebesgue-measure]

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Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
Timothy Chow's user avatar
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8 votes
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*-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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7 votes
0 answers
230 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 ...
Delio Mugnolo's user avatar
6 votes
0 answers
263 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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5 votes
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150 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
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
5 votes
0 answers
144 views

Extending gauge integral to higher dimensions/spaces and analogue of Riemann rearrangement theorem for it

The gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in $\Bbb R$, at the cost of many of the nice properties of Lebesgue integration, of which it is a ...
nimish's user avatar
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5 votes
0 answers
132 views

Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
Guy Fsone's user avatar
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5 votes
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338 views

Computing the infinite dimensional Lebesgue measure of "cubes"

There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
truebaran's user avatar
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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
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4 votes
0 answers
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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
4 votes
0 answers
285 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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4 votes
0 answers
253 views

Continuity of the Lebesgue measure w.r.t the Hausdorff metric

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff ...
Redeldio's user avatar
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283 views

trace-class embeddings

There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
Delio Mugnolo's user avatar
3 votes
1 answer
237 views

Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?

Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that \begin{equation} \int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty \end{equation} ...
Isaac's user avatar
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3 votes
0 answers
644 views

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 ...
Rob Arthan's user avatar
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
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
0 answers
71 views

Understanding simple point processes

Background I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes. A simple point process is a random variable whose ...
matteogost's user avatar
2 votes
0 answers
129 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
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2 votes
0 answers
172 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
2 votes
0 answers
54 views

Isometries and complex differentials

Assume that A is a linear operator of $L^2(D)$ onto istelf, where $D$ is the unit disk. Assume that $f\to \partial A[f](z)$ and $f\to \bar \partial A[f ](z)$ are isometries. Whether it implies that $A$...
Hengy's user avatar
  • 169
2 votes
0 answers
167 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 $A \subseteq [0, T]$ ...
RandomStudent's user avatar
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
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
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2 votes
0 answers
178 views

Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable). Also, let $f:D_1\cup D_2=D\...
Elliot's user avatar
  • 121
1 vote
0 answers
78 views

Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
Daniel Goc's user avatar
1 vote
0 answers
47 views

Understanding simple point processes (part 2)

This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods by Daley ...
matteogost's user avatar
1 vote
0 answers
196 views

Are orbits of a measurable flow always measurable with measure zero?

Let $(X, \mathcal{B})$ be a standard Borel space with a probability measure $\mu$ on $\mathcal{B}$. Let $(T_t)_{t \in \mathbb{R}}$ be a jointly measurable flow (i.e. $(T_t)_{t \in \mathbb{R}}$ is a ...
Stepan Plyushkin's user avatar
1 vote
0 answers
134 views

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
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
43 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
1 vote
0 answers
56 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
1 vote
0 answers
57 views

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 $a_n\in \mathbb{R},b_n \in \mathbb{R}^{m}, n = 1, ..., N$. ...
GreedIsGood's user avatar
1 vote
0 answers
184 views

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$...
inoc's user avatar
  • 339
1 vote
0 answers
85 views

Recursive expression of Lebesgue measure for balls with removed intersection

This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
ABIM's user avatar
  • 5,019
1 vote
0 answers
125 views

Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
user155214's user avatar
1 vote
0 answers
116 views

On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; M(|D^2u|^2)>N_1^...
MathDood's user avatar
1 vote
0 answers
301 views

Relationship between weak Lp and strong Lq topologies for q<p

Specificaly: Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence? Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
Mate Kosor's user avatar
0 votes
0 answers
44 views

Existence of derivative of distribution of exponential family?

Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
Jaimin Shah's user avatar
0 votes
0 answers
82 views

A question about associated operator on continuous functions space equiped with L2 norm

$$\text{For M a connected compact manifold, T is in }C^{1+\nu}(M,M) \text{ with } \nu\in(0,1),\\ \text{i.e. DT is some Hölder continuous function with Hölder exponent }\\ \text{, Denote m as the ...
WaoaoaoTTTT's user avatar
0 votes
0 answers
121 views

Uncountable collections of sets with positive measures

Let $X$ be a compact metric space and let $T: X \rightarrow X$ be continuous. Let $\mu$ be a $T$-invariant Borel probability measure (which we can always find by the Krylov-Bogoliubov theorem). Let $(...
Stepan Plyushkin's user avatar
0 votes
0 answers
46 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
0 votes
0 answers
204 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}...
Bogdan's user avatar
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0 votes
0 answers
81 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 ...
Bogdan's user avatar
  • 1,330
0 votes
0 answers
196 views

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,\...
Bogdan's user avatar
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0 votes
0 answers
66 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 ...
ABIM's user avatar
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0 votes
0 answers
170 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 ...
Jax989's user avatar
  • 11
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
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
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