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

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What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?

The definition of the Lebesgue measure on $\mathbb R^n$ is fundamentally tied up with the following assumption: The measure of the cartesian product of $n$ intervals should be the product of the ...
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0answers
11 views

What is the support of Radon measure on compact $T_2$ space

Q) if $X$ is a topological Hausdorff space and $\mu$ is a Radon measure, a measurable set $A$ outside the support has measure zero. How to prove it and does the above statement means ...
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2answers
47 views

Lebesgue Integral of Non-Measurable Function

In what follows I'm only considering positive real valued functions. Everywhere I look about the definition of the Lebesgue integral it is required to consider a measurable function. Why do we not ...
2
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2answers
47 views

Integrate $\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy$ using Fubini or Tonelli theorems

I am trying to show that this integral $$\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy $$ exists and is finite and then finding its value. Since $\sin \pi x$ takes ...
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1answer
24 views

Definition of the support of a measure

Having read on Wikipedia about support of measure, I found two different definitions. The first one is in terms of measure spaces "All measurable sets should have nonzero measure". The second one is ...
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1answer
23 views

Questions about complex measures

The following proposition is from the Real Analysis by Folland: where the definition of $|\nu|$ is as the following: Here is my question: In the proof of 3.13(b), $g|f|=f$ $\mu$-a.e. ...
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2answers
35 views

Borel $\sigma$-Algebra definition.

Definition: The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals (a, b], where a < b in $\mathbb R$ (We also ...
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2answers
46 views

Definition of $\pi$ and d -systems.

Definition: Let $\Omega$ be a sample space. a) A d-system is a family of subsets containing $\Omega$ and closed under proper difference (if A,B $\in\mathcal D$ and A $\subseteq$ B, then B \ A ...
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1answer
29 views

exists $A \in \mathcal{F}$ such that $\mu(B\triangle A) < \epsilon$

Let $\mu$ be a probability measure on $(S, \mathcal{S})$, where $\mathcal{S} = \sigma(\mathcal{F})$ for a field $\mathcal{F}$. How do I go about showing that for each $B \in \mathcal{S}$ and $\epsilon ...
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1answer
15 views

If $(f_n)_n$ and $(g_n)_n$ converge stochastically to $f$ and $g$, then $(f_n+g_n)_n$ converges stochastically to $f+g$

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space $(E,d)$ be a separable metric space $f,g,f_n,g_n:(\Omega,\mathcal{A})\to(E,\mathcal{B}(E))$ measurable $(a_n)_{n\in\mathbb{N}}\subseteq E$ We ...
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1answer
27 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
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2answers
29 views

A probability function is determined on a dense set- Where is density used in the following proof?

A probability function is determined on a dense set- Where is density used in the following proof? Consider the following theorem and proof from Resnick's book A probability path. I cannot really see ...
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1answer
26 views

The expectation over an infinitesimal time interval

I have a deduction which I would like to formalize (I suppose with some additional measure theory): Let $N(t)\sim Pois(\lambda_t)$, where $\lambda_t$ is stochastic (we are thus looking at a Cox ...
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3answers
40 views

Difference of elements from measurable set contains open interval

Let $A\subset\mathbb{R}$ be a measurable set s.t $,m(A)>0$. Prove that the set $$B=\{x-y\mid x,y\in A\}$$contains nonempty open interval around 0. I thought to take an interval in $A$, ...
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4answers
204 views

Why is the naive recursive approach to defining Lebesgue measure not satisfactory?

As I understand it, the spirit of the Lebesgue measure is: An interval is measurable and the measure of an interval is the absolute difference between its endpoints. The measure of a countable ...
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1answer
19 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
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1answer
34 views

Question on support of measure

Let $(X,\Sigma,\tau,\mu)$ be a topological measure space, and let $K=\text{supp}(\mu)$ and $\lambda=\mu|_K$. If $M=\Sigma/N$ and $L=\Sigma_K/(N\cap K)$, where $N$ is the set of al nulls. As I think, ...
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0answers
27 views

Prove that $\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}$ [duplicate]

Let $(X,\mathcal{A}, \mu)$ be any measure space and let $u \in \bigcap_{p\in [1,\infty]} \mathcal{L}^p(\mu)$. Then $$\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}.$$ I have already proved the ...
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0answers
7 views

Definition of Bernouilli shift

Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have density $p_i$. A translation-invariant ...
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0answers
34 views

applications of Riesz representation theorem knowing the functional

Let $L_n\colon C_c(\mathbb{R})\rightarrow \mathbb{R}$ with $$ L_nf = \sum_{n=1}^{\infty}\sum_{k=0}^{n} \frac{1}{n} e^{k/n} f\left(\frac{k}{n}\right). $$ Prove that $\lim_{n\rightarrow ...
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continuity of an integral function [on hold]

Let $(X,\rho)$ be a metric space, $\mathcal{A}$ is the Borel algebra and $\mu$ is a Borel measure. Suppose $A\colon X\rightarrow \mathcal{A}$ is a lower semicontinuous function so that for each ...
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0answers
44 views

If $f_k\to f$ a.e. $\mu$-on $X$, prove that $f\in L^1(X,\mu)$ and $f_k\to f$ in $L^1$ norm.

Please help me solving this problem: Suppose $(X,\mathcal{A},\mu)$ is a finite measure space and $f_k$ a sequence of integrable function on $X$. Suppose further that to each $\epsilon>0$ there ...
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49 views

Inequality on length of intervals

Let $n\ge 1$ and $\{I_j\}_{j=1}^{n}$ is a set of non-degenerate subintervals of $[0,1]$. Then show that : $$ \overline\sum \dfrac{1}{|I_j\cup I_k|}\geq n^2$$ Here $\overline\sum$ denotes ...
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2answers
35 views

Fatou's lemma. Examples with limit inferior $\neq$ lim.

I have problems with understanding Fatou's Lemma. What is the reason for using $\liminf$? Can someone please give an example where $\liminf \neq \lim$. When the reason does not depend on one of the ...
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0answers
30 views

Equality of measure sets of dynamical system

This is a homework question I have been crunching my brains on for a lot of time, but unfortunately I'm stuck. I would greatly appreciate any help! The problem is as follows: We have some continuous ...
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1answer
32 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
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1answer
44 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
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2answers
47 views

The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
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1answer
16 views

Comparing marginal on product space with other measure

My previous post Unifying the treatment of discrete and continuous random variable, got successfully answered and allowed me to get further in my results. However I am facing a question that I can't ...
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2answers
30 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
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1answer
80 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated (maybe we actually need independence) random variables $\Omega\to ...
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1answer
24 views

why the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable

If $E\in\mathcal{B}$ , then the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable, where $\mathcal{B}$ is the family of Borel subsets of $\mathbb{R}$ ...
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0answers
23 views

Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
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1answer
32 views

Does $a^2P(|X |\ge a )\le EX^2 $ hold when $a<0 $?

That is, does Chebyshev's inequality hold for when $a $ is negative? I have seen some authors to require that $a $ be positive, but when Reading the proof by Rick Durrett, I cannot see that this is ...
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1answer
55 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$

Could you please help me solving this old prelim problem. Any hints are appreciated
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1answer
38 views

Boolean algebra with measures

Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic ...
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2answers
89 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
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2answers
37 views

Is every measurable set a measure-independent limit of open sets

My main question is Q1. Let $B$ be a Borel-measurable subset of $\mathbb R$. Is there a sequence of open sets $U_n$ independent of any measure such that for all Borel probability measures ...
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1answer
50 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
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1answer
50 views

Measurability of adapted processes

Let $(\Omega, \mathscr{A}, P)$ be a probability space, $(E, \mathscr{E})$ a measurable space and $X_t : \Omega \to E$, $t \geq 0$ a measurable stochastic process, i.e. the map $X : [0, \infty) \times ...
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0answers
19 views

For $E \subset \mathbb{R}$ and $\epsilon >0$, $\exists$ $(a,b)$ s.t. $\theta(E \cap (a,b)) \geq (1-\epsilon)|b-a|$ ($\theta$ Lesbegue Outer Measure)

In my notes this statement is left unproven. I want to show that for any measurable set $E \subset \mathbb{R}$ with $\theta(E)>0$, there exists an interval $(a,b)$ that covers $E$ arbitrarily ...
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1answer
82 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
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1answer
42 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
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1answer
46 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
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47 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
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2answers
33 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
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2answers
54 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
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1answer
38 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
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1answer
29 views

Showing convergence of a series almost everywhere

If $\sum_{k=1}^\infty a_k$ is convergent series of positive terms and $(\alpha_k)_{k\in \Bbb N}$ is a sequence of real numbers, then the series $$\sum_{k=1}^\infty\frac{a_k}{\sqrt{|x-\alpha_k|}}$$ ...
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2answers
32 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...