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

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Beppo-Levi: Reverse

For a merely decreasing positive sequence it fails: $$f_n:=\frac{1}{n}\chi_{[n,\infty)}:\quad\int f_n\mathrm{d}\lambda=\infty\nrightarrow0$$ For a dominated decreasing positive sequence it holds: ...
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27 views

Two possible senses of a random variable being a function of another random variable

Given two random variables X and Y (assumed measurable as usual), consider two conditions: There is a (not necessarily measurable) function $f: \mathbb R \to \mathbb R$ such that $Y = f(X)$ holds. ...
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56 views

How do I verify this fact, measure theory?

This is from my book: As you see it is up to me to verify it for myself. But I do not know how? I know that the integral of f is defined to be: $\int f d\mu=\sup\{\int sd\mu | \text{s simple,} s ...
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47 views

Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
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20 views

What can be said about a measure with given marginal measures

Let $(X,\mathcal F_X,\mu_X)$, $(Y,\mathcal F_Y,\mu_Y)$ be two measure spaces. Let $\mu$ be a measure on $\bigl(X\times Y, \sigma(\mathcal F_X \times \mathcal F_Y)\bigr)$ such that for each $A \in ...
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25 views

$f : X → \overline{\mathbb{R}}$ is measureable if and only if, $\{x|f(x) > a\}, a \in \mathbb{Q}$ is measureable.

$f : X → \overline{\mathbb{R}}$ is measureable if and only if, $\{x|f(x) > a\}, a \in \mathbb{Q}$ is measureable.With $ \overline{\mathbb{R}} = \mathbb{R} \cup \pm \{\infty\} $ It is stated in my ...
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166 views

Do there exist two singular measures whose convolution is absolutely continuous?

Let $\mu, \nu$ be finite complex measures with compact supports on the real line, and assume that they are singular with respect to the Lebesgue measure. Can their convolution $\mu\ast\nu$ have a ...
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32 views

Continuity of an abstract integral

I'm trying to prove the following: Suppose $(Y, \mathcal A, \mu)$ is a finite measure space and suppose $U \subset \mathbb R^n$ is an open set. Suppose the fuction $F:U\times Y\rightarrow \mathbb R$ ...
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53 views

Understanding the statement that $\varphi(\emptyset)=0$ implies $\varphi$ is not identically $\infty$

The proposition is from "Real and Complex Analysis" by Rudin.It states: Let $s$ be a nonnegative measurable simple function on $X$ . For $E\in\mathfrak M$ (where $\mathfrak M$ is a $\sigma$-algebra ...
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38 views

Almost surely equality

suppose that X = Y almost surely.i.e. P(X=Y)=1. Then how can one show that the events $X^{-1}(M)$and $Y^{-1}(M)$ are equal almost surely for each Borel set M ∈ B.
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Is is true that if $E|X_n - X| \to 0$ then $E[X_n] \to E[X] $?

My question is motivated by the following problem: Show that if $|X_n - X| \le Y_n$ and $E[Y_n] \to 0$ then $E[X_n] \to E[X]$. I started off by saying that since $$|X_n - X|\ge 0 $$ then $$E[|X_n - ...
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34 views

Set Theory and finite unions

Let $A$ be the collection of finite unions of sets of the form $(a,b]\cap Q$ where $-\infty\leq a<b\leq \infty$. Does $\phi\in A$?
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88 views

Premeasure and induced outer measure

Let $A \subset P(X)$ be an algebra, $A_\sigma $ the collection of countable unions of sets in $A$, and $A_{\sigma \delta}$ the collection of countable intersections of sets in $A_\sigma $. Let $\mu_0 ...
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40 views

How could I recreate the proof of the Dominated Convergence Theorem?

I saw a proof of the Dominated Convergence Theorem that goes like this: If $X_n \to X$, $|X_n| \le Y $, and $E[Y] < \infty$, prove that $E[X_n] \to E[X]$. First, define $Z_n = X_n + Y$. Then, ...
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31 views

Is the set $\{(\omega, r) : f(\omega) > r\}$ measurable?

Let $(\Omega,\Sigma)$ be a measurable space and let $(\mathbb{R}, \mathcal{B})$ be the standard 1-dimensional Borel space. Let $f: (\Omega, \Sigma) \rightarrow (\mathbb{R}, \mathcal{B})$ be a ...
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44 views

Need help with some equivalent statements of measurability [duplicate]

I want to know why the above statements are true. Thank you!
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32 views

Every measure of natural numbers and the power of natural numbers as their sigma algebra looks like this…

Let X= $ \mathbb{N} $ ans S= P($ \mathbb{N} $) . Prove that every measure $\mu $ in $(X,\mathcal S)$ can be obtained by an unique non-negative extended sequence of real numbers $(a_{n})$ as follows ...
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19 views

Measurability using a premeasure

Let $X$ be a set, $\mathcal{A}$ be an algebra on $X$, and $l$ be a premeasure on $\mathcal{A}$ such that $l(X)< \infty$. Let $\mu^{*}$ be the outer measure generated by $l$. We wish to show that a ...
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1answer
40 views

How to prove $EX_n\uparrow EX$?

How to prove $EX_n\uparrow EX$? The question is as follows. If $EX_1^- < \infty$ and $X_n \uparrow X$, then $EX_n \uparrow EX$. Maybe using monotone convergence theorem, but I really have no ...
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48 views

How to prove this expectation equality?

How to prove this expectation equality? I am studying probability theory by myself and I find it hard. Thanks!
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32 views

Countable additive of a measure

Suppose we have a field of sets $\mathcal F$ such that no infinite union of members of $\mathcal F$ belong to it. Let $m$ be any finitely additive measure on $\mathcal F$, then $m$ is ...
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19 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called ...
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30 views

Question about Measure theory. The least $\sigma$- algebra generated by certain subsets of $\mathbb{R}$ equals the Borel set of $\mathbb{R}$.

I would appreciate any help with the following exercise: Consider the collection $F_0$ of subsets of the real numbers that can be written as a finite union of disjoint intervals of type: $(a,b]$: ...
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104 views

Exercise 30 from Chapter 1 (“Measure Theory”) of Stein and Shakarchi's “Real Analysis”

Consider the following exercise from [1] (p. 44): 30 If $E$ and $F$ are measurable, and $m(E) > 0$, $M(F) > 0$, prove that $$ E + F = \{x + y : x \in E, x \in F\} $$ contains an ...
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109 views

How to prove $E(\sum\limits_{n=1}^\infty X_n)=\sum\limits_{n=1}^\infty EX_n$

How to prove that if $X_n>0$, then $E(\sum\limits_{n=1}^\infty X_n)=\sum\limits_{n=1}^\infty EX_n$? I think I should use something like monotone convergence theorem, but I really don't know how to ...
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1answer
54 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
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2answers
50 views

How to prove the above expectation inequality?

If $\mathbb{E}[|X|^k]<\infty$ then for $0<j<k$, $\mathbb{E}[|X|^j]<\infty$, and furthermore $\mathbb{E}[|X|^j]\leq(\mathbb{E}[|X|^k])^{j/k}.$ How to prove the above expectation ...
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178 views

When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?

Consider two measure spaces $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$, where $\mu\times\nu(X\times Y)>0$. Given two measurable functions $f:X\to \mathbb{R}$ and $g:Y\to\mathbb{R}$ such that ...
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1answer
45 views

Jordan Measure of Ascending Union

By definition, Jordan outer measure of a subset $E$ in $\mathbb{R}^n$ is the approximation to area of $E$ by finitely many open cubes(rectangles) which cover $E$. Similarly, the Jordan inner measure ...
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105 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from Introduction to Measure Theory by Terrence Tao. ...
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32 views

Borel-Cantelli Theorem

The following is a problem from Stei-Shakarchi's Real Analysis: Suppose $(E_n)$ be a countable family of measurable sets such that $\sum_n m(E_n)<\infty$. Define $E=\{ x\in\mathbb{R}^d\colon x\in ...
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29 views

Why all closed intervals of $R$ is a semi-algebra?

How the class of all closed intervals can be a semi-algebra?
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69 views

Inducing a surface area measure on $S^2$ from the Haar measure on $SO(3)$

I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example ...
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45 views

proving continuity of decreasing measurable sets, without using same results for increasing measurable sets

There is a well known result in measure theory that says that: Suppose that $(\Omega,A, \mu)$ is a measure space. If $\{E_n\}_{n=1}^\infty\subseteq A$, with $E_1 \supset E_2...$, and $\mu(E_1) ...
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1answer
60 views

Fourier coefficients of a (finite, regular, positive) measure are absolutely summable => the measure has a density

Let $\mu$ be a finite, regular, positive measure on $[0,1)$ such that $\sum_{n\in\mathbb{Z}} |\hat{\mu}(n)| < \infty$. How can I prove that there exists $f(x)$ such that $\mu(dx) = f(x)dx$? ...
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20 views

Factor theorem for $\bar {\mathcal M}(\mathcal E)^+$ (set of $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions with values in $[0,\infty]$).

Factor theorem for $\bar {\mathcal M}(\mathcal E)^+$ (set of $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions with values in $[0,\infty]$). Let $X$ be a non-empty set, let $(Y,\mathcal ...
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54 views

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring sets?

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring on sets? By a closed-open interval , I mean an interval of the form $[x,y)$ A ring of sets is a non-empty ...
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38 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
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47 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a measure space. It is well known that a measure is continuous from below as well as from above: ...
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Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
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76 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
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Counting the exact number of sets in the Borel Field generated by a collection of “unrelated” sets

Prove: The B.F. generated by n given sets "without relations among them" has $2^{(2^n)}$ members. To be perfectly clear, "without relations among them" means that no set in the generating ...
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87 views

Uniform Wiener-Wintner Theorem - proof

I am looking for proof of uniform version of Wiener-Wintner theorem: Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the ...
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Does this proof for the MCT hold for the extended real valued functions.

Here is a proof for the MCT, but it says that it is for the real numbers, not the extended real numbers. If we allow the function f to take the value infinity does the proof still hold? I can not see ...
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22 views

Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
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Prove $\int s d \mu = \sum^n_{j=1} a_j \mu(A_j)$ for $s=\sum^n_{j=1} a_j 1_{A_j}$ not a standard representation of $s$.

Let $(X, \mathcal E, \mu)$ be a measure space. Let $s \in \mathcal S\mathcal M(\mathcal E)^+$ be a simple function written as $s= \sum^n_{j=1} a_j 1_{A_j}$ , $a_j \ge 0, A_j \in \mathcal E$. Prove ...
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35 views

Condition for a function $f: \mathbb R \rightarrow \mathbb R$ being right or left-continuous at $a \in \mathbb R$.

I know that $f: X \rightarrow \mathbb C$ is continuous if and only if for every convergent sequence $(x_n)$ in $X$ the identity holds $\lim_{n \rightarrow \infty} f(x_n) = f(\lim_{n \rightarrow ...
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30 views

THE sigma-ring or A sigma-ring?

I have two questions about sigma-rings and measure spaces. Let $(\Omega, \mathscr{F}, \mu)$ be any measure space. Is $\mathscr{F}$ THE sigma-ring of this space or A sigma-ring of this space? If ...
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28 views

Measure theory: proof of the “Standardproof” given theorem.

Measure theory: proof of the "Standardproof" given theorem. Let $(X, \mathcal E)$ be a measurable space. Let $W \subseteq \mathcal M(\mathcal E)$ (set of measurable $\mathcal E$-$\mathcal ...
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42 views

How to do you compute the probability a record occurring in a sequence of independent experiments?

Consider a sequence of independent experiments, each of which produces a random integer in N with the probability mass function ${p_k}$. The pmf is the same for all the experiments and also strictly ...