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

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2
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1answer
43 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 ...
5
votes
1answer
327 views

Folland, Chapter 1 Problem 17

Problem 17: If $\mu^*$ is an outer measure on $X$ and $\{A_i\}_{i=1}^{\infty}$ is a sequence of disjoint $\mu^*$-measurable sets, then $\mu^*(E\cap \cup_{j=1}^{\infty} A_j)=\sum_{j=1}^{\infty}(E\cap ...
-3
votes
0answers
16 views

About derivative function

I have a tricky, If f is continuous on [a,b]. Please show me a derivative function g on [a,b] which is satisfied : i) f(x)
4
votes
1answer
265 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
3
votes
3answers
98 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 ...
2
votes
1answer
30 views

Are there Cantor sets of non-zero measure?

A cantor set is generated by removing a centered open interval from $[0, 1]$ and repeating the process infinitely on the two leftover segments. What if the first interval we remove is of length $r$, ...
0
votes
0answers
27 views

Intersection of a measurable set an interval

Suppose that $E$ is a Lebesgue-measurable set, with $m(E)>0$. Prove by contradiction that there exists an open interval $I$ such that $m(E \cap I)>0.99m(I)$. I'm really struggling in this ...
4
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0answers
44 views
+50

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 ...
1
vote
1answer
26 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
0
votes
1answer
40 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 ...
2
votes
1answer
11 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. ...
2
votes
0answers
22 views

pseudo inverse of a finite-to-one continuous map and measurability

Given that $\pi: X \to Y$ is a continuous onto map between compact metric spaces such that the fiber $\pi^{-1}(y)$ is a finite subset of $X$ for all $y$, is the map $y \mapsto \pi^{-1}(y)$ guaranteed ...
0
votes
1answer
40 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 ...
0
votes
0answers
21 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. ...
0
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0answers
28 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
0
votes
0answers
12 views

Premeasure in measure theory

Let $A$ be the collection of finite unions of sets of the form $(a,b]\cap Q$ where $-\infty\leq a<b\leq \infty$.Prove the following. (a)$A$ is an algebra on $Q$. (b)The $\sigma-$algebra generated ...
0
votes
0answers
19 views

Measurable simple function

Statement: Given a measurable simple function $\phi$ on $[a,b]$ , there is a step function $g$ on $[a,b]$ such that $g(x) = \phi(x)$ on $[a,b]$ except for a set of measure $ < \in/3$. If $ m \leq ...
1
vote
1answer
16 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 \, \, \, \, \, ...
0
votes
1answer
21 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 ...
1
vote
0answers
10 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 ...
2
votes
2answers
33 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 ...
0
votes
1answer
26 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 ...
2
votes
1answer
37 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 $\mu(dx) = f(x)dx$? ...
1
vote
2answers
48 views

A problem in Sigma algebra

I'm looking for ideas to solve the following problem: Let $(X,\mathbf{X})$ be a measurable space. If the $\sigma$-algebra $\mathbf{X}$ consists of a infinite number of subsets of $X$, then ...
0
votes
0answers
26 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 ...
-1
votes
0answers
17 views

outer measure in measure theory

Let $\mu$ be a finite measure on $(X,M)$ and let $\mu^*$ be the outer measure induced by $\mu$. Suppose that $E\subset X$ satisfies $\mu^*(E)=\mu^*(X)$(but not that $E\in M$). If $A,B\in M$ and $A\cap ...
0
votes
0answers
22 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 $F:U\times Y\rightarrow \mathbb R$ ...
1
vote
1answer
20 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.
2
votes
2answers
41 views

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 - ...
0
votes
1answer
20 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 $\phi\in A$?
0
votes
0answers
17 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 ...
1
vote
0answers
20 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, ...
0
votes
2answers
28 views

Need help with some equivalent statements of measurability [duplicate]

I want to know why the above statements are true. Thank you!
1
vote
1answer
18 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 ...
1
vote
0answers
12 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 ...
1
vote
1answer
28 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 ...
2
votes
1answer
30 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!
1
vote
1answer
18 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 ...
1
vote
0answers
10 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 ...
0
votes
1answer
19 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: $(a,b]$: ...
2
votes
3answers
68 views

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

How to prove that if $X_n>0$, then $E(\sum\limits_{i=1}^\infty X_n)=\sum\limits_{i=1}^\infty EX_n$? I think I should use something like monotone convergence theorem, but I really don't know how to ...
3
votes
1answer
43 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 ...
-1
votes
1answer
33 views

How to prove $EX_n \uparrow EX$? [on hold]

How to prove that if $EX^-<\infty$ and $X_n \uparrow X$ then $EX_n \uparrow EX$? Can someone help me with this problem? Thanks so much!
0
votes
1answer
38 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!
1
vote
3answers
31 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 ...
11
votes
3answers
389 views

Probability of selecting a non-measurable set

If you randomly select a subset of $[0,1]$, what is the probability that it will be measurable? Edit: This question may be unanswerable as asked. If additional assumptions could be made to make it ...
0
votes
1answer
22 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 ...
1
vote
2answers
105 views

How is every subset of real numbers measurable despite the existence of a non-measurable set?

We know the existence of nonmeasurable subsets of $\mathbb R$ by Vitali, but many books and lecture notes on real analysis still include the statement that every subset of real numbers is measurable. ...
0
votes
0answers
18 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 ...
0
votes
0answers
13 views

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

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