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

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Prove $\mu(\{x:f(x)>t\})=m(\{s>0:f^*(s)>t\})$ for every $t>0.$

Let $f$ be positive measurable function on space $X$ with $\sigma$ finite measure $\mu$ for which $\mu (\{x:f(x)>t\})<+\infty$ for every $t>0$. Define $f^* (s)=sup\{s\geq ...
2
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1answer
29 views

If $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu),$ then $\varphi \in L^\infty$

Let $\varphi$ be a measurable function for which $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu).$ Show that $\varphi \in L^\infty(\mu).$
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1answer
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Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
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1answer
12 views

What is meaning of symbol $\wedge$ in Probability with Martingales by Williams

On page 62 of probability with Martingales by Williams, he defines: For $n \in \mathbb{N}$, define $X_n(\omega) := \{ |Y(\omega)| \wedge n\}^p$ I know $\wedge$ in the context of set theory, ...
3
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1answer
22 views

$\int f = \lim\int f$ but $\int_{E}f\neq\lim\int_{E} f_{n}$

This is exercise 2.13 in Folland's Real Analysis textbook Let $(X, \mathcal{M})$ be a measurable space. Suppose $\{f_{n}\}\subset L^{+}$, $f_{n}\to f$ pointwise, and $\int f=\lim\int ...
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2answers
23 views

Measures: Sigma-Additivity vs. Continuity

Let $R$ be a ring of sets that contains the empty set and $\mu$ be a positive and finite set function on $R$. If $\mu$ is countable additive, then it is continuous from below and above: $$A_n\uparrow ...
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Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$

Let $A \subset \mathbb R^n$, prove that there is $H$: $A \subset H$, with $H$ a $G_{\delta}$ set such that $|A|_e=|H|$. The definition of $|A|_e$ is $|A|_e=\inf\{m(U): A \subset U\}$ where the ...
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2answers
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Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
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Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
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2answers
55 views

Probability of events in an infinite, independent coin-toss space

I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the ...
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1answer
40 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
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1answer
19 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
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1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
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0answers
22 views

Pi-System Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Given a probability space. Let $\mathfrak{I}_1, \mathfrak{I}_2, \mathfrak{I}_3$ be $\pi$-systems on $\Omega$ such that for k ...
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2answers
33 views

$L^{\infty} (X, \mu)$ is not separable? [on hold]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
2
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1answer
16 views

Does simply-connected imply measureable?

The famous examples of non-connected sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space is ...
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1answer
27 views

open set $O$ such that $\partial(\overline{O})$ has positive measure

Find an open set $O$ such that $\partial(\overline{O})$ has positive measure. The hint is to consider a Cantor set, with positive measure. But that does not work, because all the Cantors are closed ...
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1answer
17 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
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0answers
26 views

Borel image of the projection [on hold]

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps Borel sets to Borel sets?
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1answer
128 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
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1answer
33 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
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2answers
25 views

A proposition about positive random variables and expected values

I have problems to give a proof for the following proposition: Consider a random variable $X$ with values in $[0,+\infty]$. If $P(X=+\infty)>0$, then $E(X)=+\infty$ (notation: $E(X)=\int X ...
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2answers
217 views

Is this set measurable?

Let $E$ be a subset of $\mathbb{R}$. Assume that $\forall x\in E, x$ is a limit point of $E\setminus\{x\}$. Then, is $E$ Lebesgue-measurable? For example, any perfect subset, open subset or ...
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1answer
52 views

Confusion with real numbers and random variables; Integration and Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let $X_n$ be iid RVs with the same continuous dist function. Let $E_1 = \Omega$ and for $n \geq 2, E_n = (X_n > X_m ...
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1answer
62 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
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1answer
23 views

Probability of highest common factor in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let s > 1 and let $\zeta(s) = \sum_{n=1}^{\infty} {n^{-s}}$. Let X and Y be independent $\mathbb{N}$-valued random variables ...
5
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1answer
52 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
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38 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
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Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
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0answers
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Completely stumped on exercise cooncerning the characterisation of Jordan measure - would anyone be willing to give a hint?

In Terry Tao's notes on measure theory he has the following exercise, I have no idea how to deal with the last statement, I would really appreciate it if someone could give a hint for the final case. ...
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1answer
20 views

Conditions for “forward” measure-preservation

A transformation $T$ being $\mu$-invariant is by definition a transformation satisfying $$\mu(T^{-1} E) = \mu(E)$$ for all measurable sets $E$. I was wondering what are sufficient conditions for being ...
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30 views

Independence Exercise in Rosenthal

In Rosenthal's, "A First Look At Rigorous Probability Theory", $\exists$ this exercise: Exercise 3.6.19. Let $A_1,\ A_2,\ldots$ be independent events. Let $Y$ be a  random variable which is ...
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4answers
97 views

Convergence in measure implies pointwise convergence?

In showing that we can replace pointwise convergence with convergence in measure in the Lebesgue Dominated Convergence Theorem, I made the following claim: 1.) $f_n\to f$ in measure ...
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1answer
36 views

Proof that $X$ countable, $\mathcal M$ algebra on $X$ implies $\mathcal M$ a $\sigma$-algebra

In my measure theory class, I believe the professor made the claim that if $X$ was a countable or finite set and $\mathcal M$ was an algebra on $X$, then $ \mathcal M$ was a $\sigma$-algebra. I am ...
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1answer
28 views

Independence of Events in Rosenthal

$\exists$ this exercise in Rosenthal's A First Look at Rigorous Probability Theory: For letters d and e, how do you show that the ff events are independent? My attempt: It suffices to show that ...
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1answer
24 views

Question on a variation of Borel Cantelli Lemma

In this question, what is the purpose of the summation? If the limit of the sequence is zero, the corresponding series is convergent. Does the desired conclusion not then follow from BC1?
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1answer
31 views

Continuity of measure and integration

Suppose that f is a measurable function $(\Omega, \mathfrak{F}, \mu)$ such that $\int_{A}f \, d\mu \geq 0 \forall A \in \mathfrak{F}$. Prove that $f \geq 0 \ \mu$-almost surely. Hint: Let $A_n = ...
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1answer
45 views

Using Borel-Cantelli Lemma

Let $X_1, X_2,\ldots$ be iid Geometric(p) where $p \in (0,1)$. Thus if $q=1-p$, then $P(X_n > k) = q^k$ for $k\geq 0$. Prove that for any fixed $\epsilon \in (0,1)$, $P(X_n > ...
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Possion integral for measure dominated by the maximal function of the measure [on hold]

Recently, I was thinking a problem about Possion integral for measure dominated by the maximal function of the measure, that is to say, let $\mu$ be a regular Borel measure in $\mathbb{R}$, define its ...
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2answers
57 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
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1answer
37 views

Do the subsets of $\mathbb N$ that have asymptotic density form an algebra?

Consider $ \Omega=\mathbb{N}.$ Is said that a $E\subset\mathbb{N}$ has a density limit if the following limit exists: $$\rho(E)=\displaystyle \lim_{n\rightarrow \infty} \dfrac{\#\{k\in E: k\leq ...
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1answer
56 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
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1answer
27 views

“Structure of a measure space is the coarsest among all substantial structures on a set…”

In the book Lectures and Exercises on Functional Analysis by Helemskii I have stumbled upon the following note: The Rohlin theorem and similar results (see e.g., [19],[20]) show that the structure ...
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20 views

Absolute continuity for non-measures?

Let $B$ be the collection of Borel subsets of $R^2$. A measure on $B$ is said to be absolutely continuous with respect to area if any subset with area 0 has measure 0. Is there a natural ...
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1answer
15 views

Absolute continuity of two-dimensional measures

Absolute continuity has two different meanings: one for functions and one for measures. The Wikipedia page explains the relation between the two notions in the following way: A finite measure μ ...
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1answer
22 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
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Determine the Lebesgue measure of the set of numbers without a fixed digit in its decimal expansion [closed]

Let $A \subset [0,1]$ be the set of all numbers without 9 over its decimal expansion. Determine the Lebesgue measure of $A$.
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+50

Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1} = \{x \in \mathbb{R}^d:|x|=1\}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every rotation $r$ of $\mathbb{R}^d$ and ...
2
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1answer
37 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...