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

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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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1answer
13 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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2answers
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Need help with some equivalent statements of measurability

I want to know why the above statements are true. Thank you!
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1answer
12 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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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
21 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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1answer
24 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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1answer
15 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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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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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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1answer
32 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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3answers
64 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 ...
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1answer
32 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!
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1answer
36 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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3answers
29 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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1answer
27 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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0answers
9 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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17 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. Show that an unsigned function $f: \mathbb{R}^d \to [0, +\infty]$ is a simple ...
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2answers
18 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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0answers
11 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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32 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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1answer
18 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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0answers
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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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1answer
13 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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12 views

Extension of Premeasures

Here, a premeasure is a countably additive set function whereas a measure is one acting on a sigma-algebra. Not every positive premeasure admits an extension to a positive measure as the following ...
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1answer
34 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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1answer
26 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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2answers
16 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...
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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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1answer
20 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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22 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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1answer
18 views

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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1answer
12 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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1answer
23 views

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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31 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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1answer
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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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1answer
21 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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1answer
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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 ...
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Can simple functions take the value infinity?

I don't think my book is clear about this. It is "a course in real analysis", by weiss. Now I am in the chapter about the general lebesge integral, and we are going to develop the non-negative ...
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25 views

The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in ...
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About counting measure on Borel sets

Let $\mathcal{C}$ be all finite unions of half open intervals, $\mathcal{A}=\sigma(\mathcal{C})$, i.e., the Borel $\sigma$-algebra. Suppose that $\mu$ is the counting measure, and $\nu=2\mu$. Can ...
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1answer
30 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
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1answer
54 views

Is this description of “sigma-algebra generated by collection of subsets” right?

Disclaimer: sorry for my poor english and edition. Claim: If $M\subseteq \mathcal{P}(X)$, then $\Sigma(M)=M_3$, where: $\Sigma(M)$ is the sigma-algebra generated by $M$ $M_1=\{A\subseteq X:(A\in ...
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3answers
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How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
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Help me in this excercise with Hermetian scalar product. [on hold]

On the vector space $C[-1,1]$ is the Hermetian scalar product $(f,g):=\int_{-1}^{1}f(x)\overline{g}(x)dx$ defined. a)Determine the function in W = span{1,x}, closest to $f(x)=x^3$ is. b)Determine ...
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1answer
21 views

Measure defined in an atypical way

I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ). Let $\Omega \subset ...
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1answer
39 views

Borel measurable functions

Suppose that f:R→R is a Borel measurable function and let h:R^2→R be defined by h(x,y)=f(x)+f(y). Prove that h is Borel measurable
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1answer
31 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
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1answer
15 views

How does one prove that elements of the Borel set are regular?

How does one prove that elements of the Borel set are regular? A Borel set of course being any element of the Borel sigma algebra (say A), and regular meaning that for a given real number e, there is ...