Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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69 views

union of sigma algebra is sigma algebra

I need help to demonstrate: Let $F,G$ two $\sigma$-algebra in $\Omega$, if $F\cup G$ is $\sigma$-algebra in $\Omega$, show $F\subset G$ or $G\subset F$. Thanks
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1answer
27 views

Convergence of sequence of smooth functions

I have the following $\{f_n\}^\infty$ sequence of smooth functions where $f_n:[0,1] \to \Re$ and $f_n(0) = 0$ with the following assumptions: $$ f_n(x) \to f(x)\ \forall x \in [0,1] $$ $$ f_n' \to g ...
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1answer
84 views

Conditioning a conditional probability to a sigma algebra

Suppose I have two random variables, $X$ and $Y$, defined on the space $(\Omega,\mathcal{F},P_1)$ which can both take the values $0,1,\ldots,N$. Suppose further I want to define the probability of ...
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1answer
23 views

Can anyone help me understand one step on the proof of Fatou's lemma?

Let $\{f_n\} \to f$ pointwise on $E$, then $\int_E f \leq \liminf \int_E f_n$. The book claims that it suffices to show that if $h$ is any bounded measurable function of finite support for which ...
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0answers
25 views

With $\lambda^*$ as the Lebesgue outer measure, $\epsilon\in(0,1),\ \lambda^*(E)>0$, find interval $I$ s.t. $\lambda^*(E\cap I)>\epsilon\lambda^*(I)$

We're to show that some interval $I$ satisfies the condition in the title. I.e., there exists an interval $I$ such that $\lambda^*(E\cap I)>\epsilon\lambda^*(I)$. I know that because any interval ...
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0answers
14 views

Divergence test for a double integral $\int \int |f| dxdy$

lets say $\int (\int f) dxdy \ne \int (\int f) dydx $ can we conclude $\int \int |f| dxdy$ diverge? $f$ is assumed to be measurable over $x,y$ and $(x,y)$.
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1answer
21 views

Conditions where $\mu$ is semifinite and where $\mu$ is $\sigma$-finite

This comes out of the book Real Analysis by Folland: $\mu$ is semifinite if and only if $f(x) < \infty$ for every $x\in X$, and $\mu$ is $\sigma$-finite if and only if $\mu$ is semifinite and ...
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2answers
38 views

almost everywhere Vs. almost sure

I'm reading a book about measure theory and probability (first chapter of Durret's Probability book), and it's starting to switch between the terms "a.e." and "a.s." in different contexts. I'm ...
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1answer
24 views

Can anyone explain one step in the proof of Fatou's lemma?

Let $\{f_n\} \to f$ pointwise on $E$, then $\int_E f \leq \liminf \int_E f_n$. The book claims that it suffices to show that if $h$ is any bounded measurable function of finite support for which ...
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0answers
45 views

reference request for $L^p(\partial\Omega)$ in real analysis textbooks

Let $\Omega$ be a bounded open set in $\mathbb{R}^d$. Would anybody come up with a real analysis textbook which contains detailed introductory treatment of the space $L^p(\partial\Omega)$?
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26 views

Can anyone explain one step in the proof Egoroff' Theorem?

Assume E has finite measure and let $\{f_n\}$ be a sequence of measurable functions on E that converges pointwise to $f$. Define $E_n = \{ x \in E \ \|\ |f(x) - f_k(x)| < \eta \quad \forall k ...
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0answers
18 views

Bounded measurable function is a uniform limit of simple functions

I want to show that if $f : \mathbb{R}^{d} \to [0, \infty]$ is a bounded unsigned measurable function if and only if $f$ is the uniform limit of bounded simple functions. I know how to do this if $f$ ...
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1answer
13 views

Application of Holder and Poincare inequality

Let $p,q >1$ and $u \in W^{1,p}_{0}(\Omega)$ and $v \in W^{1,q}_{0}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ with smooth boundary. Suppose that $p,q \in (1,N)$, $q^{'} \in ...
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0answers
33 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...
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1answer
54 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the functiona ...
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2answers
31 views

Understanding the $\sigma$-algebra of a sum of random variables

I've been studying discrete martingale theory and I have been wondering about the relationship between $\sigma\{X+Y\}$, and $\sigma\{X\}$ and $\sigma\{Y\}$ for two random variables X and Y. Is it ...
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0answers
58 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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2answers
41 views

Why is the line $\mathbb{R}$ a null set in the plane $\mathbb{R}^2$

According to Wikipedia, the straight line $\mathbb{R}$ is a null set in $\mathbb{R}^2$. That means, the line $\mathbb{R}$ can be contained in $\bigcup_{k=1}^\infty B_k$, where $B_k$ are open disks ...
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1answer
31 views

Uniform convergence of monotone bounded functions

Let $(\mathbb{R}, \mathcal{B}, \mu)$ be a measurable space. Let $f: \mathbb{R} \to \mathbb{R}$ be monotone non-constant measurable function and $\exists a \exists b\forall x:a < f(x) < b$. Let ...
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5answers
399 views

What is wrong in this proof: That $\mathbb{R}$ has measure zero

Consider $\mathbb{Q}$ which is countable, we may enumerate $\mathbb{Q}=\{q_1, q_2, \dots\}$. For each rational number $q_k$, cover it by an open interval $I_k$ centered at $q_k$ with radius ...
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0answers
30 views

Prove that if $f$ is measurable, then $f(Tx)$ is measurable.

Definition of measurability of function $f$ is said to be measurable if $\{x:f(x)>a\}$ is measurable. Prove that, for $f$ defined and measurable in $\mathbb{R}^n$, $f(Tx)$ is measurable, where ...
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1answer
97 views

Does Fubini Theorem hold when one space is infinite dimensional?

Can we exchange the order of integration when one of the integration is over infinite dimensional space? This is related to my previous question here: Optimize over measure on function space . Let ...
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1answer
56 views

How to prove that if $f$ is continuous a.e., then it is measurable.

Definition of simple function $f$ is said to be a simple function if $f$ can be written as $$f(\mathbf{x}) = \sum_{k=1}^{N} a_{_k} \chi_{_{E_k}}(\mathbf{x})$$ where $\{a_{_k}\}$ are distinct values ...
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2answers
83 views

Book about intuition behind Lebesgue measure

I recently completed a course in Real analysis covering Lebesgue and Borel measure, Fourier series, $L^p$ spaces and such. I can solve problems in these topics but am afraid that I do not truly ...
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1answer
17 views

Formula for the characterisc function of a infinite intersections of sets.

Let any characteristic function on set $S$ be denoted by $\mathcal{X}_S$. Note that if $E\cap F=A$ and $E\cup F=B$, then $$ \mathcal{X}_A=\mathcal{X}_E\mathcal{X}_F\hskip 0.4cm \text{and}\hskip0.4cm ...
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2answers
34 views

Why is a pointwise limit of a measurable function measurable?

I was reading the proof on Royden, but could not convince myself of the following line. The Union and intersections do not quite make sense to me. Can anyone give me some intuition and help me ...
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1answer
27 views

Do these integrals converge to 0?

Assume that you have a probability space $(\Omega,\mathcal{F},P)$. And you have a positive random variable $Z$, with $E[Z] =1$. You can then define a new probability space $(\Omega,\mathcal{F},P')$, ...
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5 views

Lebesgue Outer measure equivalence: |E\A|_e

Let $E\subset \mathbb {R^n}$ a measurable set and $A\subset E$. Prove $|E\setminus A|_e=inf \{|E\setminus F|: F\subset A, F \text{ closed} \}$. ($|•|$ is the Lebesgue's measure) $\leq$ is obvious ...
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26 views

Optimize over measure on function space

I'm an absolute newbie in analysis, so this might be a dumb question. Let $S$ the space of non-negative, monotone functions from R to R. Is the following optimization problem well-defined? ...
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0answers
16 views

Relation between Kolmogorov Zero-One Law and Random Graphs Zero-One Laws?

I know of Zero One laws for Random graphs (such as those concerning monotonic or first-order-logic properties). I also know about Kolmogorov's zero one law for tail sigma algebras. Apart from the ...
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1answer
59 views

Is this proof correct? (Showing that if $f$ is zero except on a closed set $E$ of null measure then $f$ is integrable

Let $E \subset I\times I$, where $I = [0,1]$, and suppose that $E$ has null measure and is closed. Then I want to prove that a bounded function $f : I\times I \to \mathbb{R}$ that is null except on ...
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1answer
40 views

Are probability measures weak-* closed?

Non-duplicates This is in a different setting, and this only deals with compact spaces which is the easy case. Now for the question. Let $X$ be a locally compact Hausdorff space. $\mathcal{C}_0(X)$ ...
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0answers
41 views

Regularity of a measure $n(E) = \int_{E} f(x) dx$

I would like to show that the (positive) measure $n$ on $\Bbb R \setminus \{0\}$ defined by $n(E) = \displaystyle \int_E \frac{dx}{|x|}$ is outer regular ($dx$ being the usual Lebesgue measure). ...
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25 views

Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$

I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it. Proposition: $\lim \inf ...
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37 views

Is there a better way to understand set operation

In measure theory, for example, we always need to find a good way to express a set to show its measurability. eg: we write $ A \cap B$ as $$ A \cap B = D\backslash( \ (D\backslash A) \cup ...
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31 views

Proof that $\mathcal L^2 \supsetneq \mathfrak B^2$.

Let $\mathfrak L^d$ be the $\sigma$ -algebra of all Lebesgue-measurable subsets and $\mathfrak B^d$ the one of the Borel sets in $\mathbb R^d$. I want to prove that $\mathfrak L^2 \supsetneq \mathfrak ...
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42 views

The set of fixed points of a Borel function

Let $f: \mathbb R \rightarrow \mathbb R$ be a Borel function, is the set $\{x : f(x)=x\}$ Borel? Edit: As short questions seem to be quite unpopular here, I'll elaborate a little: As the graph of the ...
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1answer
37 views

Limit of $n-1$ measure of the boundary of a sphere

The measure of a sphere of radius $R$ centered in $0_{\mathbb{R}^n}$ in $\mathbb{R}^n$ is \begin{array}{l l}\int_{B_0(R)}dx_1\ldots dx_n & =\int_0^R\rho^{n-1}d\rho ...
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0answers
52 views

$E\subseteq \mathbb{R}$ and $m^*(E)<\infty$: $E$ is measurable iff $m^*(E)=m^*(A)+m^*(E-A)$ for any $A \subseteq E$

Suppose $E\subseteq \mathbb{R}$ and $m^*(E)<\infty$. Prove that $E$ is measurable if and only if for any subset $A \subseteq E$, we have: $$m^*(E)=m^*(A)+m^*(E-A)$$ If $E$ is measurable and $A$ is ...
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1answer
15 views

Sequence of measurable sets inequality

Let $(E_n)_{n \in \mathbb N}$ be a sequence of measurable sets in $\mathbb R^m$ and $k \in \mathbb N$. Show that if $G=\{x \in E_n \text{for at least k values of n}\}$, then $G$ is measurable and ...
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0answers
7 views

Set difference produces a regular measure

Let $\mu$ be a regular measure on $\mathbb R^n$ and $S\subset\mathbb R^n$ countable. Is the measure $\tilde\mu(A)=\mu(A\setminus S)$ also regular? I feel like the answer should be yes but I can't ...
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2answers
58 views

Are these sets Borel-measurable?

Are the following sets Borel-measurable and if so, what is the value of the measure? 1) A = {(x,y) ∈ $[0,1]^2$| x and y rational} 2) B = {(x,y) ∈ $[0,1]^2$ | x or y rational} 3) C = {(x,y) ∈ ...
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1answer
36 views

How to show that $m^*(A \cup B) + m^*(A \cap B) \leq m^*(A)+m^*(B)$ for any $A,B \subseteq \mathbb{R}$?

How to show that $m^*(A \cup B) + m^*(A \cap B) \leq m^*(A)+m^*(B)$ for any $A,B \subseteq \mathbb{R}$. I first thought that I could easily prove it by using the sub-additivity property of the ...
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38 views

Why does this function not converge in measure?

Definition Let $f$ and $\{f_k\}$ be measurable functions which are defined and finite almost everywhere in a set $E$. Then $\{f_k\}$ is said to converge in measure on $E$ to $f$ if for every ...
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21 views

Almost sure convergence on conditional probability

Say we have a compact space $X$ and a probability measure $P$ on $X$. Assume that we know that for some event $A$ the sequence $f_n(x)\rightarrow f_\infty(x)$ converges a.s. in $Q$ which is the ...
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1answer
21 views

limitation of unions of sets

$\mathbb N$ is the positive integer, for $A\subset \mathbb N$, define $$p(A)=\lim_{n\to\infty}\frac{1}{n}\#\{j\in A:1\le j\le n\}$$ if the limit exists. $\#$ is the number of elements of the set ...
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1answer
23 views

Measurable functions in $(N, P(N))$

Given $P(N)$ is the power set of $N$ - set of natural numbers, and $h$ is a counting measure. Prove that every functions $g: N\rightarrow R$ is $h$-measurable. My attempt: First, it's trivial that ...
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0answers
31 views

A curve with Lebesgue measure non zero

In this Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ...
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0answers
30 views

Countable union in Borel sigma algebra

If Borel sigma algebra is defined to be the sigma algebra generated by the class of all open intervals of $\Omega$=(0, 1], then all these open intervals will be uncountable. My question is how this ...
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1answer
26 views

Lebesgue–Radon–Nikodym Theorem Explanation

From Folland, the theorem is as follows: The Lebsgue–Radon–Nikodym Theorem Let $\nu$ be a $\sigma$-finite signed measure and $\mu$ a $\sigma$-finite positive measure on $(X,\mathcal{M})$. There ...