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

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Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
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1answer
39 views

Problem 2.2 Real Analysis Folland

I want to solve this problem: If $f,g:X\to \overline{\mathbb R}$ are measurable and $c$ is any extended real value the function $$h(x)=\begin{cases}c& \text{if}\ f(x)+g(x)\ \text{is undefined}\\ ...
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1answer
29 views

What is the measure zero of uncountable set .

Recently I was reading Methods of Real Analysis by Goldberg and had the following question. 7.1 Corollary: Every countable subset of $\mathbb{R}$ has measure zero. How can we describe the ...
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29 views

The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as ...
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3answers
49 views

Real Analysis, Folland Problem 1.3.11 Measures

Background information - Let $X$ be a set well equipped with a $\sigma$-algebra $M$. A measure on $M$ (or on $(X,M)$ or on $X$ if $M$ is understood) is a function $\mu: M \rightarrow [0,\infty]$ such ...
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15 views

Modulus of Integral defines a measure?

Say you have an integrable function $f: \mathbb{R}^n \to \mathbb{R}$ which does change sign. Does the set function $$m(A):=\left| \int_A f \text{d} x\right|$$ define a either a measure or a ...
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1answer
21 views

On the interpretation of the limsup of a sequence of events

In the context of understanding Borel-Cantelli lemmas, I have come across the expression for a sequence of events $\{E_n\}$: $$\bigcap_{n=1}^\infty \bigcup_{k\geq n}^\infty E_k$$ following Wikipedia ...
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19 views

Measurability and integrability of set and function

My textbook said: Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure. if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given ...
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1answer
25 views

$f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. Find $\lim \int f_n d\lambda$.

Let $X = \mathbb R$, $\textbf{X} = \textbf{B}$ and $\lambda$ the Lebesgue measure on $\textbf{X}$. I have the following: $f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. I need to find the following: ...
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1answer
34 views

Sequence of Events - Basic understanding

I have an idea of the meaning of a sample space, and the events included in a sigma algebra. However, I am stuck in the definition of a sequence of events. My difficulty is in the fact that in a ...
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46 views

Find the Lebesgue measure of the following sets.

Find the Lebesgue measure of the following sets: i) A=$(\cup_{n=1}^\infty [2^n, 2^n + \frac{1}{2^n}))$ \ $\mathbb{Z}$ ii) B=$(\cup_{n=1}^\infty (n^n, n^n + \frac{1}{2^n}))$ $\cap$ $\mathbb{Q}$. For ...
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0answers
12 views

Given an event field, is there a random variable generating it? [duplicate]

In probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, for any event field $\mathcal{G}\subset\mathcal{F}$, there always exists a random variable $X$, such that $\sigma(X)=\mathcal{G}$? Is ...
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20 views

Conditional Distributions vs. Stochastic Processes

Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ...
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50 views

Sigma-algebra on $\Omega = \{1,2,3,4,5\}$ generated by $\epsilon = \{\{1,2,4,5\},\{2,3\}\}$

Let $\Omega = \{1,2,3,4,5\}$. Let $\epsilon = \{\{1,2,4,5\},\{2,3\}\}$. Find $\sigma$ ($\epsilon$), generated by ($\epsilon$), and justify answer. Could someone please give me some direction and ...
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2answers
25 views

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$.

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$. I have never done an example of convergence of sequences that have characteristic (indicator) ...
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2answers
37 views

Real Analysis, Folland Problem 1.3.14 [duplicate]

If $\mu$ is semifinite measure and $\mu(E) = \infty$, for any $C > 0$ there exists $F\subset E$ with $C < \mu(F) < \infty$. Attempted proof - Suppose $E\in M$ with $\mu(E) = \infty$ then ...
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1answer
40 views

Can the unit interval be the disjoint union of countably many “super-dense” parts?

I'm curious about this question in the case where $f$ is not necessarily measurable. I think what it comes down to is this: Is there an $\varepsilon<1$ and a partition of $[0,1]$ in countably ...
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1answer
55 views

Decomposition of complex Radon measures

Suppose you have a complex Radon measure $\mu$, treated as a distribution. Then does every such Radon measure admit a decomposition of the form $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
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24 views

Is there a there a non intersecting mapping to unit square.

Is there a way to go from the fat cantor set to a half unit square in a non intersecting way using Hilberts curve? How would I go about constructing a non intersecting space filling curve of non zero ...
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1answer
48 views

Is every set of measure zero countable?

I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?
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1answer
22 views

Exercise 10.N of The elements of integration and Lebesgue measure Bartle's book

If $a_{mn}\ge 0$ for $m,n\in\mathbb{N}$, then $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}(\le +\infty).$
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Prove that $\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$ is measurable.

If I have two measurable functions $f:\mathbb R^p\to\mathbb R^m$ and $f:\mathbb R^q\to\mathbb R^n$, how can I prove that $$\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$$ is ...
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1answer
30 views

A question on atoms in measure theory.

The definition in Bogachev's book goes as follows: Now let $\mathcal{A}$ be a $\sigma$ algebra and let $\mu$ be a finite countably additive measure. 1.12.7 Definition. The set $A\in \mathcal{A}$ ...
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1answer
31 views

A finitely additive measure is a measure if and only if we have continuity from below

A finitely additive measure $\mu$ is a measure if and only if it is continuous from below. I want to know how I should proceed in proving this statement. My idea is to first assume we have a ...
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3answers
71 views

$f $ vanishes iff $f$ integrable.

Can someone give me a sketch for proving this: Let $f:\mathbb R\to\mathbb R$ be a monotone and therefore measurable function. Show that $f$ is integrable if and only if $f(x)=0$. Some hints would be ...
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1answer
60 views

Real Analysis Folland, Problem 1.3.8 Measures

If $(X,M,\mu)$ is a measure space and $\{E_j\}_{1}^{\infty}\subset M$, then $\mu(\liminf E_j) \leq \liminf \mu(E_j)$. Attempted proof - Let $(X,M,\mu)$ be a measure space and ...
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4answers
70 views

$\int_\Omega f d\mu = 0 $ if and only if $f(x)=0$ almost everywhere

can someone give me a hint on what kind of theorem/definition I should make use of to solve this? Let $(\Omega,\mathfrak A, \mu)$ be a measure space and $f:\Omega \to \mathbb R$ a non-negative ...
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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 ...
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1answer
27 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 ...
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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. ...
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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 ...
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3answers
474 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 ...
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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 ...
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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 ...
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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 ...
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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
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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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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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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
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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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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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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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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
53 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 ...