The lebesgue-measure tag has no usage guidance.

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### 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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**1**answer

98 views

### Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [on hold]

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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vote

**1**answer

79 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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22 views

### Q: The relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure [migrated]

Q:how we can discribe relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure?

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45 views

### A usage of mean value theorem for Nemytskii operators

Let $F$ be a real-valued continuously differentiable function over $\mathbb{R}$. Let $\Omega$ be a bounded set in $\mathbb{R}^2$; and $w_1$ and $w_2$ be in $H^1_0(\Omega)$. I am doing this calculation ...

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**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 ...

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**5**answers

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### 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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### 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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241 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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156 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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**1**answer

88 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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**1**answer

76 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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46 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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**1**answer

74 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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207 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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129 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
$...

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**1**answer

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

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64 views

### A Vitali–Hahn–Saks type theorem for the Lebesgue-Stieltjes vector measure

Let
$a,b\in\mathbb R$ with $a<b$
$I:=(a,b]$
$E$ be a $\mathbb R$-Banach space
$\operatorname{BV}(\overline I,E):=\left\{g:\overline I\to E\mid g\text{ is of bounded variation}\right\}$
If $g\in\...

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135 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 ...

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223 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 ...

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411 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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186 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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133 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 ...

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121 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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309 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 $...

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266 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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890 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 ...

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95 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 ...

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133 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 $A\subset \mathbb{R}^n$ ...

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269 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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255 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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72 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 ...

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**1**answer

263 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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**2**answers

236 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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### 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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589 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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126 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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### 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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### 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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**1**answer

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### 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,...

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### $\forall g\in L^2(\Omega)$ 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 ...

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### 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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### 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 ...

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**1**answer

109 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 ...

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143 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 ...

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872 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?

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### 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\...

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### Can ITTM recognize a non-measurable set?

Throughout the question ITTM refers to Hamkins' infinite Turing machines, though I will be interested in results related to stronger models.
Recently I was wondering, is it consistent that there is ...

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**1**answer

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### Which domain maximizes the energy of the Lebesgue measure?

This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...

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537 views

### Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?
For ...