## New answers tagged measure-theory

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 ...

- 21

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 ...

- 201k

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:...

- 11.4k

2
votes

Accepted

### 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\...

- 5,172

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 $\...

- 347

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 ...

- 91

2
votes

Accepted

### 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 ...

- 30k

1
vote

Accepted

### 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 ...

- 19.8k

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 ...

- 1,121

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 $\...

- 85.1k

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=\...

- 23.9k

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}(\...

- 23.9k

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 ...

- 85.1k

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 ...

- 23.9k

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)

- 1,808

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
$...

- 339

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 ...

- 85.1k

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 ...

- 177

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(\...

- 85.1k

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 $\mu$...

- 38.5k

Top 50 recent answers are included

#### Related Tags

measure-theory × 2591pr.probability × 707

fa.functional-analysis × 602

real-analysis × 459

reference-request × 244

geometric-measure-theory × 224

gn.general-topology × 219

set-theory × 182

stochastic-processes × 128

probability-distributions × 128

integration × 110

ergodic-theory × 109

ca.classical-analysis-and-odes × 103

lebesgue-measure × 98

descriptive-set-theory × 96

banach-spaces × 77

ds.dynamical-systems × 75

ap.analysis-of-pdes × 69

mg.metric-geometry × 56

harmonic-analysis × 56

gr.group-theory × 48

borel-sets × 47

limits-and-convergence × 46

lo.logic × 45

haar-measure × 45