Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

Filter by
Sorted by
Tagged with
0 votes
0 answers
144 views

Remainder-balancedness of primes

Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
Dominic van der Zypen's user avatar
0 votes
1 answer
174 views

Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let $X := \mathbb R^n$, $C_b(X)$ the space of all real-valued bounded continuous, $C_c(X)$ the space of all real-valued continuous functions with compact supports, and $C_c^\infty(X)$ the space of ...
Analyst's user avatar
  • 647
6 votes
1 answer
270 views

A characterisation of continuous real functions

Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
Nate River's user avatar
  • 4,822
0 votes
1 answer
90 views

Sums of powers of measures of $p$-adic balls

Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
Daniel Loughran's user avatar
7 votes
2 answers
165 views

Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$? I was ...
Saúl RM's user avatar
  • 7,916
6 votes
2 answers
152 views

When can we extend a function on a $\lambda$-system to a probability measure?

Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is, (i) $\Omega \in \mathcal{L}$, (ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \...
Jason Swanson's user avatar
2 votes
1 answer
179 views

References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent): ...
FD_bfa's user avatar
  • 147
2 votes
3 answers
337 views

More natural example of measurable but not progressive process

All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean. Consider ...
tsnao's user avatar
  • 490
2 votes
0 answers
114 views

A technical question concerning convolution product

Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$. Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
Guy Fsone's user avatar
  • 1,033
6 votes
1 answer
178 views

Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?

Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e., ...
Akira's user avatar
  • 815
1 vote
1 answer
360 views

Motivation for Ionescu-Tulcea extension theorem (as opposed to Kolmogorov's)

I recently asked a question on the differences between Ionescu-Tulcea and Kolmogorov extension theorems (ITET and KET for short). A lot of my confusion has been cleared there and what I understood ...
tsnao's user avatar
  • 490
0 votes
1 answer
68 views

Meyer's example of a separable process with no path regularity

This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link. In the following excerpt from ...
tsnao's user avatar
  • 490
17 votes
1 answer
1k views

Does the set of square numbers adhere to Benford's law in every base?

Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to Benford's law for the first digit in every base $b\geq 2$? Precise formulation of what it means for a set $T\subseteq \omega$ to "...
Dominic van der Zypen's user avatar
5 votes
2 answers
544 views

Kolmogorov vs Ionescu-Tulcea extension theorem (again)

Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there. I've recently ...
tsnao's user avatar
  • 490
3 votes
3 answers
364 views

Progressive measurability intuition from Bichteler's *Stochastic integration with jumps* book

In the Stochastic Integration with Jumps Bichteler gives a very intuitive definition of progressive measurability I've never seen before: Although I like this intuition very much, I cannot find a ...
tsnao's user avatar
  • 490
4 votes
1 answer
520 views

Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
Akira's user avatar
  • 815
1 vote
1 answer
55 views

Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable?

Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$. $f:X \to \overline{\mathbb R}$ is called $\mu$-...
Akira's user avatar
  • 815
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
  • 7,916
4 votes
1 answer
339 views

Does Szemerédi's theorem hold for sets with positive upper Banach density?

We say that a set of natural numbers $A\subseteq \omega$ has positive upper density if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$ Szeméredi's theorem states that every $A\subseteq \omega$ ...
Dominic van der Zypen's user avatar
0 votes
1 answer
200 views

Upper density versus upper Banach density on $\omega$

For $A\subseteq\omega$ we define the upper density by $$d_u(A) = \lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1}.$$ For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ ...
Dominic van der Zypen's user avatar
2 votes
0 answers
112 views

Measure on the places of $\bar{\mathbb Q}$

Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...
manifold's user avatar
  • 299
8 votes
2 answers
920 views

Is there a measure theory for proper classes?

This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes? Of course when one tries to define measures on "large sets" ...
aduh's user avatar
  • 839
2 votes
1 answer
314 views

Textbook definition for "path measure" or "probability measure over paths"

I need a formal definition for the path measure for stochastic differential equations. Which textbook or paper should I consult?
can't stop me now's user avatar
3 votes
1 answer
204 views

A level set of non-constant real analytic function

Assume that $u: B\to [0,1]$, where $B$ is an open ball in $\mathbb{R}^n$, is a nonconstant real analytic function and let $t\in[0,1]$ and let $\mu(t)=\operatorname{Volume}(\{x\in B: u(x)>t\})$. Why ...
Koha's user avatar
  • 31
3 votes
2 answers
235 views

Is there something like a "self-avoiding Markov chain" on a continuous space?

If stumbled accross self-avoiding walks. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants. However, as far as I can see they are ...
0xbadf00d's user avatar
  • 161
1 vote
0 answers
74 views

Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space

Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...
Nik Bren's user avatar
  • 499
1 vote
1 answer
88 views

Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?

Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
Akira's user avatar
  • 815
1 vote
0 answers
39 views

Does the constrained Wasserstein barycenter admit a blue noise property?

Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
0xbadf00d's user avatar
  • 161
2 votes
0 answers
101 views

A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$

Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
Analyst's user avatar
  • 647
0 votes
0 answers
129 views

Approximate range of Radon-Nikodym derivative in a dynamical system

Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...
Sanae Kochiya's user avatar
4 votes
1 answer
264 views

Supremum of infimum of measure of members of a free ultrafilter

For a set $A\subseteq \omega$ we let the upper density of $A$ be defined as $d^+(A) := \lim\sup_{n\to\infty}\frac{|A\cap(n+1)|}{n+1}$. Let $\text{FrU}(\omega)$ be the collection of free ultrafilters ...
Dominic van der Zypen's user avatar
3 votes
1 answer
187 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
4 votes
2 answers
192 views

Is the conditional expectation of a Caratheodory function a Caratheodory function?

Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
Condor5's user avatar
  • 165
1 vote
0 answers
156 views

Wiener Integral and its distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field. Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
heppoko_taroh's user avatar
2 votes
0 answers
125 views

What are examples of infinite-dimensional Banach spaces that are also measure spaces?

I am interested in examples of infinite-dimensional vector spaces that are Banach spaces or even Hilbert spaces are measure spaces Instead of the full vector space, subsets with measure structure ...
shuhalo's user avatar
  • 4,736
2 votes
0 answers
82 views

A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged

I have recently read about about disintegration theorem, i.e., Disintegration theorem Let $X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
Akira's user avatar
  • 815
0 votes
1 answer
48 views

Symmetric and nearly additive bounded functions

Let $(y_n)_{n\ge 1}$ be a sequence with values in $(0,1)$ such that $\lim_n y_n=1$. Let also $f: [0,1]\to \mathbb{R}$ be a bounded function such that $f(0)=0$ and satisfies $$ \forall n\ge 1, \forall ...
Paolo Leonetti's user avatar
4 votes
1 answer
129 views

On partial absolute continuity

$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
Iosif Pinelis's user avatar
1 vote
1 answer
82 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

This is a follow-up to this previous question, but under stronger assumptions. Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real scalar field). Let $\tilde ...
Jochen Glueck's user avatar
1 vote
2 answers
139 views

On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition. If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)...
Uriah's user avatar
  • 13
7 votes
2 answers
379 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
Jochen Glueck's user avatar
0 votes
1 answer
103 views

If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e

I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral. Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \...
Akira's user avatar
  • 815
2 votes
0 answers
133 views

On limits of positive linear functionals

I am looking for pointers to the literature on questions of the following kind. ($Y$ and $\Omega$ might be open subsets of some Euclidean space, but I am interested in the kind of conditions that need ...
Arnold Neumaier's user avatar
1 vote
0 answers
61 views

Prescribed class of measurable sets

Let $X\neq\emptyset$ and let $\mu:P(X)\to[0,\infty]$ be an outer measure. Recall that, a set $A\subseteq X$ is $\mu$-measurable if $$ \mu(B)=\mu(A\cap B)+\mu(B\setminus A), \text{ for all }B\subseteq ...
Tatin's user avatar
  • 895
1 vote
0 answers
148 views

Generalizations of Fubini-Tonelli's theoem

Fubini-Tonelli's theorem: If $(X, A, \mu)$ and $(Y, B, \nu)$ are $\sigma$-finite measure spaces and $f: X\times Y \to [0,\infty]$ is a measurable function, then $$ \int _{X}\left(\int _{Y}f(x,y)\,{\...
mathqf's user avatar
  • 11
1 vote
1 answer
146 views

Show that a certain convergence of measures is equivalent to a certain convergence of integrals

Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of a measures. We know, by the Portmanteau Theorem, that: $$\int f d \mu_n \to \int f d\mu, \quad \forall \, f \in C_b \hbox{(class of continuous and ...
PSE's user avatar
  • 13
2 votes
1 answer
539 views

A question about the proof of the Levy-Khintchine representation Theorem

I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy. In the proof of the "only if" part of the Levy-Khintchine ...
MAOC's user avatar
  • 123
0 votes
1 answer
244 views

Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?

I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
Akira's user avatar
  • 815
2 votes
1 answer
161 views

Joint irreducibility and aperiodicity of two independent Markov chains

Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
Dasherman's user avatar
  • 203
1 vote
1 answer
227 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
Analyst's user avatar
  • 647

1
5 6
7
8 9
58