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2 votes

Proof of Krylov-Bogoliubov theorem

If you still need a reference: You can find a clear exposition of the result and the proof of the theorem in "Ergodic Theory" by Einsiedler 2013 (free PDF here). Or see the original work by ...
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7 votes
Accepted

Supremum of infimum of measure of members of a free ultrafilter

The answer is: zero. The reason is that every ultrafilter has zero as the infimum of the upper density of its members. To see this, observe that if a set $U$ is in the ultrafilter $\mathcal{U}$, with ...
0 votes

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

Here is a positive answer for the case that $\Sigma_0$ is generated by a random variable with values in a Polish space, so that we can use regular conditional probabilities and for some kernel $\kappa:...
2 votes
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Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Firstly, a set $P$ of positive measure need not contain anything of the form $A\times B$, for example consider for some $k\in\mathbb{R}\setminus\{0\}$ the set $P=\{(x,y)\in\mathbb{R}^2;x-ky\not\in\...
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1 vote

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

Edit: The below answer is invalid, since $\Sigma_0$ is a sub sigma algebra of $Y$, not $X \times Y$. I think the answer is no - take $X = Y = [0, 1]$, and $f(y, x) =y$. Pick some Borel set $E \subset [...
  • 1,729
1 vote

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

This is a characterization of $\sigma$-finiteness: If $h$ is an $L^1$-function with values in $(0,\infty)$ then $A_n=\{h\ge 1/n\}$ are measurable sets with $\bigcup_{n\in\mathbb N} A_n=\Omega$ and $\...
1 vote
Accepted

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

Below is my formalization of @Nik's hints to finish the proof. Let's prove that $$ \sum_{m=1}^M \|H_m\|^q_{L_{p}(\mu_m, X)^*} \le \|H\|^q_{L_{p}(\mu, X)^*} \quad \forall M \in \mathbb N^*. $$ Let $\...
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3 votes
Accepted

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

It is always true that $L_q(\mu,X^*)\hookrightarrow L_p(\mu,X)^*$ isometrically (without the assumption that $X$ has the Radon-Nikodym property). We need the Radon-Nikodym property to guarantee that ...
2 votes
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell_2$ that are of the form $\sum_n a_n f_n$, where $f_n = e_n + e_{n+1}$, $(e_n)$ is the ...
1 vote
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Symmetric and nearly additive bounded functions

Let $C$ be the usual Cantor set in $[0,1]$, and split the complement $I\setminus C$ as a disjoint union of intervals $I_1,I_2,\dots$. Let $L_i$ and $R_i$ denote the left and right half of $I_i$, and ...
2 votes
Accepted

On partial absolute continuity

The notation $L^2$ for Lebesgue measure is confusing. I denote $\lambda$ and $\lambda_2$ the Lebesgue measure on $\mathbb{R}$ and $\mathbb{R}^2$, respectively. The answer is no. Fix a discrete measure ...
2 votes

Joint irreducibility and aperiodicity of two independent Markov chains

$\newcommand{\X}{\mathcal X}\newcommand{\Y}{\mathcal Y}\newcommand{\si}{\sigma}\newcommand{\B}{\mathscr B}$Since $\X$ and $\Y$ are Polish spaces, the corresponding Borel $\si$-algebras $\B(\X)$ and $\...
0 votes
Accepted

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

It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly $$ \int_{\mathbb{R}}hd\mu=\...
6 votes

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

I believe the following is a simple counterexample: Let $(X,\mu)$ be $\mathbb{N}$ with the counting measure (so I will be writing $\ell^2$ for $L^2(X,\mu)$. Let $g=0$ and $\tilde g(n) = (-1)^n$. Let ...
  • 23.9k
6 votes
Accepted

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

The answer is no and the following result provides a quite interesting counterexample. This is a known result, but I am not sure where to find it in the literature. Theorem. If $f\in L^1_{\rm loc}(\...
1 vote
Accepted

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

$\newcommand\R{\mathbb R}\newcommand\ip[1]{\langle #1 \rangle}$In these notes, two related definitions of truncation functions are given. In Definition 5.6, a truncation function is defined as a ...
4 votes
Accepted

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

It seems that this can be deduced from the Portmanteau theorem: Assume the convergence of the intergals $\int fd\mu_n$ for all $f\in C_b$ vanishing in a neighbourhood of $0$ and fix $E\in C_\mu$ with $...
3 votes

Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...
2 votes
Accepted

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

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941). (Alternative link at Project Euclid)
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0 votes
Accepted

Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts

I represent below @NikWeaver's idea of improving $\frac{1}{2} [\cdot]' \le [\cdot]$ to get $\frac{1}{\sqrt 2} [\cdot]' \le [\cdot]$. For complex number $z = x + iy$ with $x,y \in \mathbb R$, we have $...
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0 votes

Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

No, the measure density condition is not necessary I would say. Possibly there are more precise arguments, but I would argue as follows, in a nutshell: The desired inequality can be proven using ...
  • 1,808
2 votes
Accepted

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$By the polar ...
2 votes

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

First we state here some general results about Radon measures following Bichteler, K. Integration: A Functional Approach, Birkhauser, 1991 and Bichteler, K. Integration theory: With Special Attention ...
4 votes
Accepted

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\...
4 votes
Accepted

Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?

An example. $X = [0,1]$ with the usual metric. $\mathcal C[0,1] = \mathcal C_b[0,1] = \mathcal C_0[0,1]$. $\mathcal C[0,1]^* = \mathcal M[0,1]$. Let $\mu_n$ be the unit point-mass at $1/n$ and $\mu$...
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