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

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2
votes
1answer
49 views

$\int_{A} fdxdy=0 $ For every rectangle A with area 1. Then is it f=0 a.e? [closed]

Is it true that the function $f \colon \mathbb R^2 \to \mathbb R$ satisfying condition $\int_{A} f \,\textrm{d}x \,\textrm{d}y=0$ for every rectangle $A$ whose area is $1$ must be identically 0 ...
1
vote
1answer
26 views

What is the Difference Between a Version and a Modification of a Stochastic Process?

Under what circumstances would one say that: The stochastic process $X$ is a version of the stochastic process $Y$? Background: See here for a related but slightly different question on ...
0
votes
0answers
23 views

Can anyone explain one step in the proof of the Lebesgue theorem?

If the function f is monotone on the open interval (a, b), then it is differentiable almost everywhere on (a, b). Proof: Assume f is increasing. Furthermore, assume (a, b) is bounded. ...
2
votes
1answer
45 views

Can anyone explain one claim about the infinity norm?

Define $||f||_\infty $ as the infimum of the essential upper bounds for f. For each natural number n, there is a subset $E_n$ such that $|f| \leq||f||_\infty + \frac{1}{n}$ on $E - E_n$ and ...
6
votes
3answers
472 views

What does “the average continuous function is nowhere monotonic” mean?

I plan on asking my professor what he meant by "average continuous function," but as it is possible that this is a concept as vague as the statement, I was hoping to get some interesting ...
0
votes
1answer
32 views

Radon-Nikodym derivative as a Martingale

Let $(\Omega,\mathscr{F}, P)$ be a probability space, let $\nu$ be a finite measure on $\mathscr{F}$, and let $\mathscr{F}_{1}$, $\mathscr{F}_{2}$,... be a non-decreasing sequence of $\sigma$-fields ...
2
votes
1answer
35 views

Can anyone explain one sentence in the proof of the claim that monotone functions are continuous except at a countable number of points?

Claim (page 108 of Royden's Real Analysis): Let f be monotone on the open interval (a,b). Then f is continuous except at a countable number of points. In the proof, the book states that assume ...
1
vote
0answers
20 views

Question on construction of Product Measure,

As part of constructing product measure one shows that the measure of a rectangle is well defined. The standard way to do this is to show that no matter how the rectangle is written, its measure is ...
1
vote
1answer
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
1
vote
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 ...
2
votes
1answer
82 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 ...
0
votes
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 ...
1
vote
0answers
24 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 ...
0
votes
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 $(x,y)$.
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
44 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 $L^p(\partial\Omega)$?
0
votes
2answers
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 ...
0
votes
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 $f$ ...
0
votes
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 ...
1
vote
0answers
32 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, ...
3
votes
1answer
47 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 function ...
0
votes
2answers
25 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 ...
1
vote
0answers
41 views
+50

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 ...
0
votes
2answers
40 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 ...
0
votes
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 ...
21
votes
4answers
329 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 ...
2
votes
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 ...
0
votes
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 ...
2
votes
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 ...
5
votes
2answers
78 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 $(\Omega,\mathcal{F},P')$, ...
0
votes
0answers
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 ...
1
vote
1answer
17 views

Kantorovich-Rubinstein Theorem - lower semi-continuity of distance function

For instance in "Topics in Optimal Transportation" by Cédric Villani, it is claimed that the 1-Wasserstein-distance of two measures $\mu, \nu$ on a certain space $X$ can be expressed as the supremum ...
0
votes
0answers
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? ...
0
votes
0answers
15 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 ...
1
vote
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 ...
2
votes
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. $\mathcal{C}_0(X)$ ...
1
vote
0answers
40 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). ...
1
vote
0answers
24 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 ...
0
votes
0answers
36 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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
3
votes
1answer
29 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
6 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 ...