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

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0
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1answer
144 views

Two question on ternary Cantor set & Jordan content

Is it true that all subsets of the Cantor set have Jordan content zero? What is the definition of countably generated Boolean algebra? Does the Boolean algebra of subsets $[0,1]$ which ...
4
votes
1answer
55 views

Why is $\mu(E)=0$?

(Ergoden theorem) Let $(\Omega,\mathcal{A},\mu,T)$ be an ergodic dynamical system and $f\in L_{\mu}^1$. Then $$ \lim_n \frac{1}{n}\sum_{k=0}^{n-1}f\circ T^k=\int f\, d\mu~~\text{a.s.} $$ ...
1
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3answers
45 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...
4
votes
1answer
119 views

What is a motivation for this theorem and what is an example this theorem is applied?

If one doesn't know a motivation, it's hard to memorize such a theorem. So do I. Rudin RCA p.30 Let $(X,\Sigma,\mu)$ be a measure space such that $\mu(X)<\infty$ and $f\in L^1(\mu)$. ...
2
votes
1answer
391 views

Outer and inner approximation of set with finite outer measure

I was wondering if somebody could help me out with a solution for the following problem (taken from Royden's Real Analysis, 4e. (ch. 2.4, prob. 18): Let $E$ have finite outer measure. Show that ...
6
votes
1answer
70 views

Under what conditions on $f$ does $\|f\|_r = \|f\|_s$ for $0 < r < s < \infty$.

Question: If $f$ is a complex measurable function on $X$, such that $\mu(X) = 1$, and $\|f\|_{\infty} \neq 0$ when can we say that $\|f\|_r = \|f||_s$ given $0 < r < s \le \infty$? What I ...
3
votes
1answer
103 views

Is there a Lebesgue sum-like definition for a Bochner integral?

For a Lebesgue integral, I generally see two equivalent definitions of the integral of a function $f:X\rightarrow\mathbb{R}$. One is based on Lebesgue sum $$ ...
2
votes
1answer
41 views

Question concerning the proof of the Ergodentheorem by Birkhoff

Let $(\Omega,\mathcal{A},\mu,T)$ be an ergodic dynamical system and $f\in L_{\mu}^1$. Then it is a.s. $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^k=\int f\, d\mu. $$ Now I ...
0
votes
1answer
54 views

Under what conditions do all singletons belong to sigma algebra?

Let's suppose that (A, F) is a measurable space (A underlying set, F the sigma algebra), and that F arises from some topology T. I would like a theorem of the form: All singletons belong to F if and ...
2
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0answers
149 views

How to define a probability distribution over a function space?

What is the mathematically rigorous way of defining a probability distribution over some function space e.g. $L^1[0,1]$? Edit: After reading about the basics of measure theory, I realized that the ...
1
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1answer
122 views

Is the following intersection of a set and a $\sigma$-algebra also a $\sigma$-algebra

Consider the statistical experiment of a dice roll. A probability space with $\left\{\Omega, \mathscr{F}, P\right\}$. Now $\Omega =\{1,2,3,4,5,6\}$, and $\mathscr{F}$ is the $\sigma$-algebra ...
2
votes
1answer
28 views

$\DeclareMathOperator{\esssup}{ess sup}\sup_{m \geq n} \, \esssup |u_n - u_m| = \esssup \, \sup_{m \geq n} |u_n - u_m|$

Let $u_n, u \in L^{\infty}(E)$ and let $u_n \rightarrow u$ almost everywhere. Is it true that $$ \DeclareMathOperator{\esssup}{ess sup} \sup_{m \geq n} \, \esssup_E |u_n - u_m| = \esssup_E \, \sup_{m ...
0
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0answers
35 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
1
vote
2answers
355 views

How do I evaluate the Lebesgue measure of a ball?

I remember I saw a post related to this question somewhere here, but I cannot find this now. How do I evaluate the Lebesgue measure of a ball? What I only remember is that the proof used the gamma ...
3
votes
1answer
81 views

Measurability of the set of differentiable function in the borel sigma algebra of the continuous functions

I was studying stochastic calculus and it came up this question of measurability: Let $\mathcal{C}=\mathcal{C}([0,1])$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ with the uniform ...
0
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0answers
55 views

Lebesgue Diffrentiation

Let f be lebesgue integrable in $[a,b]$ (with respect to lebesgue measure of course). We want to show that there exsist an antidrivative function $F$ which is absolutely continuous. On my try to show ...
1
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0answers
94 views

Circle rotation (dynamic system)

Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto ...
4
votes
1answer
102 views

Measure theory, probability, tail events.

I have a problem with tail events. At the top of page 19 of Stefan Grosskinsky's lecture notes, it is pointed out that $A := \{\omega: \lim_{n\rightarrow\infty}X_n(\omega) \text{ exists}\}$ is a tail ...
2
votes
0answers
70 views

Prove the theorem of Egoroff

(Theorem of Egoroff.) Let $(\Omega,\mathfrak{A},\mu)$ be a measure space with finite mesaure $\mu$. Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real-valued, measurable functions that ...
1
vote
1answer
79 views

Jensen's inequality for countable probability space

One form of Jensen's inequality for the finite case, tells us that $$ \sum_{x \in X} p(x) \log q(x) \leq \log\sum_{x \in X} p(x) \cdot q(x) $$ For positive p(x), and $\sum_{x \in X} p(x) = 1$, ...
1
vote
1answer
19 views

Set equality involving supremum of functions

Consider a sequence of measurable functions $f_n$. I would like to know if the following statements are true or false. For a positive number $c$, \begin{align}\{\sup_{n\in \mathbb{N}} |f_n|>c\} = ...
0
votes
2answers
45 views

Does there always exist a measurable function between measurable subsets of same measure?

This is a just a curiosity question that randomly popped in my head. Let $(X,\mathcal{M},\mu)$ be a measure space. Let $A$ and $B$ be two measurable subsets of $X$ with $\mu(A)=\mu(B)$. Do we know ...
5
votes
1answer
209 views

Borel $\sigma$-algebras on the Skorohod space $D[0,1]$

On the Skorohod space $D[0,1]$ of cadlag functions one usually considers either the uniform norm $\|\cdot\|_{\infty}$ or the $J_1$-metric $\varrho$. I was wondering whether both generate the same ...
3
votes
1answer
69 views

$m(E)=0$ then $m(\lbrace x^2 : x\in E\rbrace$?

Let E be a subset of $\mathbb{R}$ with lebesgue measure zero. How can I prove that $\lbrace x^2 : x\in E\rbrace$ also has lebesgue measure zero?
1
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0answers
34 views

On a rearrangement inequality

Let us consider a measure space $(\mathbb{N},\Sigma_{\mathbb{N}},\mu)$, where $$ \Sigma_{\mathbb{N}}=\{Q\in 2^{\mathbb{N}} \colon Q \text{ is finite}\} $$ and $$ \mu(Q)=\sum_{n\in Q}n. $$ Suppose ...
0
votes
0answers
226 views

Condition Lebesgue-Beppo Levi theorem proof

I found this proof of Lebesgue-Beppo Levi theorem. I don't understaand the very last part (that with integral). Can someone clarify it to me? Theorem Proof SOURCE: Notes on Elementary Martingale ...
3
votes
3answers
890 views

Inverse image of a Borel set is a Borel set under measurable function

If I define a measurable function $f:E \to \Bbb R \cup \pm\{\infty\}$ where $E \subset \Bbb R^d$ is a measurable set as if $\forall a \in \Bbb R$, the set $$f^{-1}([-\infty,a))=\{x \in E :f(x) < ...
0
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1answer
29 views

A basic question on $L_p$ norm

How to prove that if $\mu(\omega) < \infty$ then $L_p$ norm increases to $L_\infty$ norm ?
1
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3answers
267 views

A problem about Lebesgue measurable set

I'm doing exercise in "REAL ANALYSIS" of Folland and got stuck on this problem. I got no clue on how to find the set $I$. Hope someone can help me solve this. Thanks so much Suppose $m$ is ...
3
votes
2answers
143 views

Prove that $Var(X) =\sigma^2E[N]+\mu^2Var(N)$

Given $Y_1,Y_2,Y_3......$ are iid, random variables with mean $\mu$ and variance $\sigma^2$.Suppose that N is an independent random variable taking positive integer values such that $E[N^2]$ finite. ...
1
vote
2answers
62 views

Is the function that gives you the measure of the neighborhood Borel?

Let $X$ be a compact metric space (with $\epsilon -$balls $B_{\epsilon }$) and $\mu $ a Borel probability measure. Let $a,\epsilon >0.$ Is the set $\left\{ x\in X:\mu (B_{\epsilon }(x))\geq ...
4
votes
1answer
61 views

A basic question regarding the proof of existence of product measure

Suppose that $(X,\mathcal F_1,\mu)$ and $(Y,\mathcal F_2,\nu)$ are measure spaces and suppose that $\mu$ and $\nu$ are finite measures. Define the function $\nu_E : X \to \Bbb R$ by $$\nu_E(x) = \nu ...
1
vote
1answer
73 views

Show the equi-integrability of a finite set of $\mathcal{L}_{\mu}^1$-functions

Let $(\Omega,\mathcal{A},\mu$ be a measure space. A set $\mathcal{F}$ of measurable, numerical functions is called equi-integrable if for any $\varepsilon > 0$ it exists a nonnegative, ...
1
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0answers
51 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
1
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1answer
40 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
1
vote
1answer
59 views

$f_n$ - Measurable Functions

I believe I know the answer to this question but I wanted to get someones opinion on it: let ${A_n}$ be an infinity system of sets such that $\mathbb R=\bigcup_{n=0}^{\infty}A_n$ pairwise ...
1
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0answers
38 views

Closure of algebra under countable unions

Let $X$ be some set. We define a system of sets $\mathcal{C}\subseteq \mathcal{P}(X)$ by $$\mathcal{C} = \{ S \subseteq X : \text{for all } E \subseteq X \text{ we have } \mu^*(E) = \mu^*(E \cap S) + ...
6
votes
1answer
67 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
2
votes
1answer
48 views

$\mathcal{C}$ is an algebra [duplicate]

I'm trying to show that the following set is an algebra: Let $X$ be some set. We define a system of sets $\mathcal{C}\subseteq \mathcal{P}(X)$ by $$\mathcal{C} = \{ S \subseteq X : \text{for all } E ...
0
votes
1answer
107 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
2
votes
1answer
46 views

Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...
3
votes
1answer
74 views

Is there a general way to count the number of sigma-algebras on a finite set? [duplicate]

I was asked a high-school question today which was merely asking the number of sigma-algebras on a set. Let $X$ be a set Let $S\triangleq \{\Sigma\subset P(X): \Sigma \text{ is a ...
1
vote
1answer
47 views

$C^\omega(\Omega)\cap C^\infty_0(\Omega)$.

Let $\Omega$ denote an open connected set in $\mathbf{R}$ (AKA open interval). Is it true and how can we prove it that $C^\omega(\Omega)\cap C^\infty_0(\Omega)$ consists of the zero function alone, ...
1
vote
1answer
65 views

Reference for a proof of which 2-increasing functions are joint cdf's

Can somebody give me a reference giving the detailed statement and proof of the fact that the joint cdf's of positive Borel measures $\mu$ on $\mathbb{R}^2$, so $$F(a,b) = \mu(\{(x,y) : x \leq a, y ...
1
vote
1answer
55 views

Can Haar measure fail to be bi-invariant without conjugation shrinking a set?

Let $\: \langle \hspace{-0.02 in}G,\hspace{-0.04 in}\cdot,\hspace{-0.04 in}\mathcal{T}\hspace{.02 in}\rangle \:$ be a locally compact Hausdorff topological group, let $\mu$ be a left Haar measure on ...
1
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0answers
282 views

The boundary of a compact set of positive measure has measure zero

I have a surface $M$ and a compact subset $K \subset M$ with positive Lebesgue measure. I have a certain function $\ f : K \rightarrow \mathbb{R}^n$, which I want to integrate over $K$. The problem ...
0
votes
1answer
342 views

$\sigma$-finite measure and $\sigma$-semi-finite measure

Let $ (X, \Sigma, \mu) $ it will be a space with measure. $\mu$ is $\sigma$-finite measure if it exist sequence of sets $X_{i} \in \Sigma $ and $\cup_{i=1}^{\infty}X_{i}=X$ and ...
0
votes
1answer
48 views

Mutual information of discrete and continous stochastic variable

As part of a homework, I have a "quantizer" consisting of variables $X_{1}$ and $X_{2}$ which have the following joint distribution. $X_2$ is discrete and I can assume that all probabilities are ...
1
vote
1answer
220 views

a basic doubt on continuous image of a measurable set measurable

Is continuous image (say the function is on $\Bbb R^n$) of a measurable set measurable ? Hint enough. Actually, $f: \Bbb R^n \to \Bbb R^n$ is given to be linear. I used some theorem to conclude that ...
1
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0answers
62 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...