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Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
Bogdan's user avatar
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1 vote
1 answer
42 views

Integrability in the product space can follow from a property of the Nemytskii operator?

Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
Bogdan's user avatar
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0 votes
0 answers
83 views

Integral of a measurable function with parameter is measurable?

Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that: $f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$ $f(\...
Bogdan's user avatar
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2 votes
0 answers
77 views

Question about the Nemytsky operator on $L^p$ space

Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
Bogdan's user avatar
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0 answers
43 views

Weyl equidistribution for a periodic $L^2$ function

Let $\alpha $ be a fixed irrational number. For a function $g:\Bbb R\to\Bbb C$, define $$g^*(x)=\sup_{N\geq 1} \frac{1}{N} \sum_{n=1}^N |g(x+\alpha n)| ,$$ and assume that there is a constant $C>0$ ...
blancket's user avatar
  • 189
7 votes
2 answers
429 views

Uncountable collections of distinct subsets of an interval (existence)

Throughout, $\mu$ is just the Lebesgue measure. Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
Stepan Plyushkin's user avatar
0 votes
0 answers
104 views

How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
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17 votes
1 answer
985 views

Is the pushforward of Lebesgue measure on a cube by a polynomial piecewise smooth?

Question: Let $Q \subset {\bf R}^n$ be a cube, and let $P: {\bf R}^n \to {\bf R}$ be a non-constant polynomial. Is it true that the pushforward $P_*( m\downharpoonright_{Q} )$ of Lebesgue measure on $...
Terry Tao's user avatar
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54 views

Projection measure and an integral formula for Lipschitz functions

Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as $$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
Alexander's user avatar
6 votes
1 answer
260 views

Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$

Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
David Fernandez-Breton's user avatar
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0 answers
46 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
3 votes
0 answers
150 views

Lebesgue measure of the boundary of the positivity set of a function is zero?

Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties: $w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$; $w$ is biharmonic on $C=\{w>0\}$; $w$ is subharmonic ...
Evelina Shamarova's user avatar
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0 answers
121 views

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

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

Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
Julian's user avatar
  • 113
8 votes
4 answers
676 views

Distributional derivatives are locally integrable implies the distribution is also locally integrable?

Let $T$ be a distribution on $\mathbb{R}^n$ such that there are functions $f_1,\ldots,f_n \in L^1_\text{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_j} = f_j, \forall j=1,\ldots,n. $ My ...
Jinie's user avatar
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1 vote
0 answers
82 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
2 votes
0 answers
61 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
3 votes
0 answers
80 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
3 votes
1 answer
266 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
  • 3,113
1 vote
0 answers
205 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
0 votes
0 answers
124 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
2 votes
2 answers
197 views

Limit of a integral whose integrand diverges under the limit

I am trying to simplify the following limit of integral where $\mu$ is given: $$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
user482401's user avatar
1 vote
0 answers
136 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
3 votes
1 answer
135 views

Recover an $L^1$ integrand by partial differentiation

Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by $$F(x,y) = ...
w116c576's user avatar
1 vote
1 answer
146 views

Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
  • 11
1 vote
1 answer
239 views

What is the measure of two sets which partition the reals into subsets of positive measure?

This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$. (In ...
Arbuja's user avatar
  • 1
5 votes
1 answer
373 views

Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?

Suppose $\lambda^{*}$ is the Lebesgue outer measure. Question: Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where: The range of $f$ is $\mathbb{R}$ For all real $x_1,x_2,y_1,y_2$, where $...
Arbuja's user avatar
  • 1
2 votes
1 answer
170 views

Finding an explicit & bijective function that satisfies the following properties?

Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$. Question: Does there exist an explicit ...
Arbuja's user avatar
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1 vote
1 answer
218 views

Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support

In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
CoffeeArabica's user avatar
0 votes
1 answer
78 views

Convergence in sequential Lebesgue spaces

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
Guy Fsone's user avatar
  • 1,043
-1 votes
1 answer
59 views

The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence

Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
David Gao's user avatar
  • 1,534
12 votes
3 answers
697 views

If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...
David Gao's user avatar
  • 1,534
8 votes
0 answers
217 views

*-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
  • 1,534
13 votes
1 answer
626 views

Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH

Assuming the negation of CH, let $\omega_1$ be the first uncountable ordinal, $\mathfrak{c}$ be the cardinality of the continuum. Does there exist a map $f: \omega_1 \times [0, 1] \rightarrow \...
David Gao's user avatar
  • 1,534
5 votes
0 answers
151 views

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
1 vote
1 answer
171 views

Quantitative version of Lebesgue points theorem

Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
tommy1996q'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
6 votes
1 answer
175 views

Concentration of volume towards the boundary

Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let $$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$ be the set of all ...
nullptr's user avatar
  • 93
4 votes
0 answers
315 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
  • 10.4k
26 votes
2 answers
3k views

What is the origin/history of the following very short definition of the Lebesgue integral?

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
Gro-Tsen's user avatar
  • 30.8k
3 votes
1 answer
199 views

Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
user483450's user avatar
5 votes
0 answers
152 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
6 votes
0 answers
265 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
  • 61
8 votes
1 answer
2k views

How badly can the Lebesgue differentiation theorem fail?

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that $$ \lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ? $$ This is obvious if $0$ is a Lebesgue point ...
No-one's user avatar
  • 1,077
2 votes
1 answer
209 views

Problem regarding set of positive Lebesgue measure in $\mathbb{R}^2$

Let $T_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set $$L= \bigcup_{p\in[0,1]\cap \mathbb{Q}}T_{p,1} \bigcup_{q\in[0,1]\cap \mathbb{Q}}T_{1,q} $$ and consider $P=[0,1]\times[0,...
user483450's user avatar
4 votes
1 answer
322 views

$\sigma$-algebra generated by analytic sets

The Borel $\sigma$-algebra $\cal B$ on real numbers has many good properties. For instance, all continuous functions are $\cal B/\cal B$-measurable. On the other side, not only $\cal B$ is not ...
Giafazio's user avatar
  • 185
0 votes
1 answer
174 views

$f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere

Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?
Billal Elhamza's user avatar
0 votes
0 answers
47 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
1 vote
1 answer
172 views

Topological analog of the Lusin-N property

$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets. Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
Ussesjskskns's user avatar

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