# Questions tagged [lebesgue-measure]

The lebesgue-measure tag has no usage guidance.

112
questions

**8**

votes

**1**answer

132 views

### Regarding a positive Lebesgue measure set in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure.
...

**9**

votes

**0**answers

171 views

### 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 ...

**4**

votes

**0**answers

51 views

### Measure of the boundary of an BV-exention 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-...

**3**

votes

**2**answers

78 views

### Sharp assumption for preserving Lebesgue measurability by left composition

Let $g: [0, 1] \to \mathbb R$ be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that $f \circ g$ ...

**0**

votes

**1**answer

137 views

### Problem regarding Lebesgue measure in $\mathbb{R}^2$

Let $P=A_1\times A_2,$ where $A_1,A_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue ...

**4**

votes

**0**answers

129 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 ...

**4**

votes

**0**answers

87 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 ...

**5**

votes

**2**answers

224 views

### Tight sequence of measures

This is probably a very easy question for experts in probability or measure theory.
I have a sequence of finite measures $\mu_{n}$ on a non-compact metric space $X$ such that $\mu_{n}$ converges to $\...

**2**

votes

**0**answers

53 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 $A$...

**0**

votes

**0**answers

77 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(\...

**0**

votes

**0**answers

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 ...

**12**

votes

**1**answer

162 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**

votes

**1**answer

126 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**

votes

**0**answers

49 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**

votes

**1**answer

58 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 ...

**15**

votes

**1**answer

452 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 $x \in [m_i(A), m_o(A)]$ ...

**0**

votes

**0**answers

74 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**

votes

**1**answer

518 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**

votes

**4**answers

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 ...

**-8**

votes

**1**answer

751 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**

votes

**2**answers

279 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 ...

**-1**

votes

**1**answer

137 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**

vote

**1**answer

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**

votes

**1**answer

178 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**

vote

**1**answer

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**

vote

**1**answer

352 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**

votes

**0**answers

113 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**

votes

**1**answer

165 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**

votes

**1**answer

215 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 $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...

**1**

vote

**2**answers

93 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**

votes

**0**answers

123 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\...

**7**

votes

**3**answers

567 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**

votes

**1**answer

337 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**

votes

**2**answers

712 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 ...

**1**

vote

**0**answers

314 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**

votes

**1**answer

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 ...

**-2**

votes

**1**answer

449 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**

vote

**1**answer

149 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**

votes

**1**answer

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**

votes

**5**answers

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 ...

**19**

votes

**6**answers

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**

vote

**0**answers

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**

votes

**1**answer

244 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**

vote

**1**answer

102 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**

votes

**1**answer

99 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**

votes

**0**answers

175 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**

votes

**1**answer

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**

votes

**2**answers

305 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**

votes

**0**answers

197 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**

votes

**1**answer

195 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 $B_\epsilon(a)$ ...