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

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Prove an outer measure inequality.

For each $A\subset\mathbb{R}$ we define $$\displaystyle m^*(A) = \inf \Big\{\sum_{i=1}^\infty (b_i - a_i): A\subset\bigcup_{i=1}^\infty (a_i,b_i)\Big\}.$$ Then I want to see that $\displaystyle m^*\...
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8 views

How is an adjunction map of a probability space well-defined?

The book says something like: Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $\Delta \notin \mathcal{F}$, suppose that for all $F \in \mathcal{F}$ such that $F \supset \Delta$, we have $P(...
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2answers
73 views

Prove $\liminf(M_n = -1) \subseteq ((\lim \frac{S_n}{n}) = -1)$

Let $M_1, M_2, \dots$ be iid RVs s.t. $P(M_n = n^2 - 1) = \frac{1}{n^2} = 1 - P(M_n = - 1)$. Define $S_n = \sum_{i = 1}^{n} M_i$. Prove that $P\{(\lim \frac{S_n}{n}) = -1\} = 1$. $$\sum_n P(M_n = n^...
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62 views

Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$.

Assume $E$ has finite measure and $1 \leq p < \infty$. Suppose $\{ f_n\}$ is a sequence of measurable functions that converges pointwise a.e. on $E$ to $f$. For $1 \leq p < \infty$, show that $\{...
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70 views

$f$ locally bounded, nonnegative, and measurable function integrable iff series $\int_{n=1}^{\infty}a_{n}$ converges absolutely

Suppose $f$ is a locally bounded, nonnegative, and measurable function on $[1,\infty)$ and define $\displaystyle \int_{n}^{n+1}f$, $\,\,\forall n \in \mathbb{N}$. Then, is it true that $f$ is ...
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1answer
38 views

$\sigma$-finite measure on $\mathbb{R}$ that maps half-open intervals to $\infty$

Consider $\mathbb{R}$ equipped with the Borel-$\sigma$-algebra $B$ and a measure $\mu : B \rightarrow [0, \infty]$. The measure $\mu$ is called $\sigma$-finite, if there is a sequence $A_1,A_2,...$ of ...
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201 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
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1answer
53 views

Lebesgue integral of non-negative

Assume that $f: [0,1] \rightarrow [0,\infty)$ is a Lebesgue measureable function such that $f(x) > 0$ for a.e $x.$. Show that for every $\epsilon >0$ there is $\delta >0$ such that for every ...
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1answer
37 views

Convergence of measures and potential theory

The following implication should hold: $\mu_{n}, \mu$ are positive measures whose supports are included in a compact set $K\subset \mathbb{C}$ and $$\lim_{n\to\infty}U^{\mu_n}(z)=U^{\mu}(z)$$ ...
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1answer
41 views

Integral similar to Lebesgue point theorem

Assume we are in $\mathbb R^N$ and $\Gamma$ is a ($N-1$)-rectifiable set with $\mathcal H^{N-1}(\Gamma)<\infty$ and $\mathcal H^{N-1}(\bar \Gamma\setminus \Gamma)=0$. Let $u\in C_c(\mathbb R^N)$ ...
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1answer
52 views

How to prove this inequalities involving measures

Let $(X,\mathcal M,\mu)$ be a measure space and $E_i\in\mathcal M$. Then I want to check that: 1) $\mu(\liminf E_i) \le \liminf \mu(E_i)$. 2) If $\mu(\bigcup_i E_i)<\infty$, then $\mu(\limsup E_i)...
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19 views

Borel open map and the smallest Borel sigma algebra

Set from $A$ which is a subset from $\mathbb{R^n}$ is opened if for each $ a\in A$ exist such $r \gt 0$ that exists every $x \in \mathbb{R^n}$, for which is $||x-a||\lt r$, element of set $A$. (a) ...
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1answer
33 views

Show that measure of symmetric difference $= \int_{E}\vert \chi_{A} - \chi_{B}\vert$

I have just proven that the Nikodym (pseudo)metric $\rho(A,B)$ is a pseudometric (i.e., satisfies all the metric axioms, except that $\rho(A,B)$ can be $0$ even if $A \neq B$), and now I need to show ...
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52 views

Change of variable for integration with respect to Haar measure

I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ ...
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1answer
36 views

How to show convergence of integration when domain is moving.

Let $Q\subset \mathbb R^2$ be a cube centered at $(0,0)$, with side length $2$. Let $I$ denote the segment from $(-0.5,0)$ to $(0.5,0)$. Define $$\tau(x):=\operatorname{dist}(x,I)$$ for $x\in Q$. ...
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17 views

Can a subset of $N−1$ rectifiable set be partitioned into countably many connected pieces? [duplicate]

This is a follow up question regarding to the problem discussed here. Thanks so much to @Silvia Ghinassi's nice and simple explanation there! Ok, here is the updated question: Let $\Gamma\subset \...
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2answers
46 views

The convolution of an integrable function with a $p$-integrable function is integrable

Let $\Sigma$ denote the set of Lebesgue-measurable subsets of $\mathbb{R}$, and $m$ the Lebesgue measure on $\mathbb{R}$. Let $1<p\leq \infty$, $f\in L^1(\mathbb{R},\Sigma,m)$, and $g\in L^p(\...
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1answer
26 views

Can a $N-1$ rectifiable set be partitioned into countably many connected pieces?

Let $\Gamma\subset \mathbb R^N$ be a $N-1$ rectifiable curve such that $\mathcal H^{N-1}(\Gamma)<\infty$. I am wondering that would it be possible to partition it into countably many connection ...
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1answer
25 views

“Hahn decomposition” for countable or arbitrarily many mutually singular probability measure

Let $(\mu_i)_{i\in I}$ be a collection of mutually singular probability measure on $(\Omega, \mathcal F)$.(i.e. any two of them are singular to each other). I am not sure if the following statement is ...
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30 views

Convergence in $L^p([0,1])$ of $f_n(x) = n\chi_{(\frac{1}{n+1},\frac{1}{n})} $

My attempt 1) For $1 \leq p < 2$ we have: $$ \|f_n\|_p ^p = \int_{0}^{1}{|f_n(x)|^p}dx = \int_{0}^{1}{n^p\chi_{(\frac{1}{n+1},\frac{1}{n})}}dx = \frac{n^{p-1}}{n+1} \to 0$$ Then $f_n \to 0$ ...
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1answer
33 views

Show that $f^T$ is measurable

Suppose that $F \in~L^{+}$. Show that $$ F^T = \begin{cases} 1 & \text{if }f(x)\le 1, \\[3pt] f(x) & \text{if }1<f(x) <2, \\[3pt] 2 & \text{if }f(x)\ge 2, \end{cases} $$ is ...
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2answers
44 views

Suppose $\{X_n\}$ is uncorrelated sequence,meaning $Cov(X_i,X_j)=0, i\not= j$

Suppose $\{X_n\}$ is uncorrelated sequence,meaning $$Cov(X_i,X_j)=0, i\not= j$$ If there exists a constat $c>0$ such that $Var(X_n)\leq c$ for all $n\geq 1$, then for any $\alpha > \frac{1}{2}$ ...
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1answer
50 views

Show that $F(x) = \int_{-\infty ,x}f$ is continuous on $\mathbb{R}$

Let $f \in L^1(m)$ where $m$ is the standard Lebesgue measure. Show that $$F(x) = \int_{-\infty}^{x}f \text{ is continuous on } \mathbb{R}.$$ I think what is confusing me is that my professor labeled ...
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1answer
40 views

Exercise measurable function

Let $\{f_k\}$ be a sequence of measurable functions defined on a measurable set $E$ with $m(E)<\infty$. Suppose that for each $x$ in $E$, there exist a constant $M_x$ such that $$\sup_k |f_k(x)|\...
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1answer
94 views

Measure of the irrational numbers?

I have read that the measure of the irrational numbers on an interval $[a,b] = b-a$. This both makes sense and doesn't make sense to me. If you consider that the union of the irrationals with the ...
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1answer
24 views

Is the set obtained by removing intervals of $1/4^n$ a borel set?

I have a set defined by: $D_0 = [0, 1]$. $D_1$ is obtained from D0 by removing an open interval of length $1/4$ from the middle, so $D_1 = [0, 3/8] \cup [5/8, 1]$. $D_2$ is obtained from $D_1$ by ...
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1answer
35 views

Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that : $$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$ where: $X$ is separable real Banach space. $...
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0answers
43 views

Example of a $\sigma$-additive function over a $\sigma$-algebra

While working in measure theory I've decided to study each theorem using a concrete example of concepts I deal with. In this case, I want to prove that: If $\phi$ is $\sigma$-aditive function on ...
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95 views

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very ...
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1answer
71 views

Laplace functional of sum of independent uniformly distributed random variables

I'm doing some of the exercises in Cinlar's "Probability and Stochastics" to better understand the material. This exercise (VI.1.17) is taken from page 247: Fix an integer $n \geq 1$. Let $X_1,\...
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1answer
27 views

Let $E$ be a measurable set and let $g$ be a function defined on $E$. The product $fg$ belongs to $L^1(E)$ for every function $f \in L^1(E)$ [duplicate]

Let $E$ be a measurable set and let $g$ be a function defined on $E$. The product $fg$ belongs to $L^1(E)$ for every function $f \in L^1(E)$ if and only if $g ∈ L^{\infty}(E)$.
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1answer
29 views

Extrema of a $\sigma$-additive function are attained.

I've written down a proof for the following theorem which appears in Loeve's book of probability (page 86). I believe I'm missing something because Loeve's proof is much more complicated but I can't ...
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1answer
48 views

Exercises with solutions on Elementary Measure Theory

Where can I find a nice collection of solved exercises on Elementary Measure Theory? (Rings, algebras, $\sigma$-algebras, Borel sets, measures, outer measures, Lebesgue measure, measurable functions, ...
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Given an elementary set how to obtain an open elementary set containing it whose measure is less than a desired quantity

The theorem whose proof I have a question about is: Given an elementary set $A$ of $\mathbb{R}$, i.e. a set that can be represented as at least one finite (pairwise) disjoint union of bounded ...
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1answer
43 views

Efficiently get the highest value from a function for lots of parameters. [closed]

Situation: I have a function $f$(p,q,r). This function return an integer value. I want to know the highest value returned by this function. All p,q and r can take any value between 1 to 100. So, ...
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1answer
37 views

Prove that if E is positive lebesgue measurabel set, then E − E and E + E contain non-empty open sets.

let E + E = {x + y : x, y ∈ E}, and define E − E similarly. Show that if E is a measurable subset of R of positive Lebesgue measure then E − E and E + E contain non-empty open sets. I have seen the ...
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1answer
36 views

Bounded Sequence in $L^p$ is Tight

For $1 < p < \infty$, suppose that $\{f_n\}$ is bounded in $L^p(\mathbb{R})$. Is $\{fn\}$ tight? We defined tight as : A family of measurable functions is said to be tight over $E$ provided for ...
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76 views

Sum of two families of uniform integrable random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ are two families of u.i(uniform integrable) random variables defined on the same probability space. Is $\{X_n+Y_n\}$ u.i? Proof Given $$\mathbb{E}[|X_n|\,I_{|X_n|\geq ...
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1answer
48 views

Independence and change of measure

Let $(\Omega, \mathscr F)$ be a finite probability space and $\mu$ be a probability measure on $\Omega$. Consider a sequence of random variable $\xi_n$ that are independent and identically distributed:...
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1answer
37 views

Equivalent measures and independence

Suppose that $\mu$ and $\lambda$ are two equivalent measures in a probability space $(\Omega, \mathscr F)$. Suppose that random variables $\xi_1,..,\xi_n$ in $(\Omega, \mathscr F)$ are independent ...
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1answer
71 views

Is there a borel set $A$ and a linear map $f$ such that $f(A)$ is not borel?

Is there a borel set $A\in\mathbb{R}^n$ and a linear map $f:\mathbb{R}^n\to\mathbb{R}^n$ that $f(A)$ is not borel set? I think there is but I can't find it.
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86 views

$\mathcal M(K)$ is an $\mathcal{l}_1-$sum of $L_1(\mu)$ spaces

Let $K$ be a compact Hausdorff space. I want to show $\mathcal M(K)$ is an $\mathcal{l}_1$-sum of $L_1(\mu)$ spaces, where $\mathcal M(K)$ is the dual of $C(K)$. I have got the sketch of the proof but ...
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80 views

Prove $E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$ for nonnegative random variables $X,Y$ and $p\ge0$

Suppose $X \geq 0$ and $Y \geq 0$ are random variables and that $p\geq 0$ Prove $$E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$$ Proof Since $(X+Y)^p \leq (2 \> \max\{X,Y\})^p=2^p \> \max \{X^p,Y^p\}\...
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58 views

The intersection of all events in a sequence has probability $\lim \limits _{k \to \infty} P(A_k)$

If a sequence $A_1, A_2, A_3, \dots$ of events is decreasing, show that the intersection of all events in the sequence has probability: $\lim \limits _{k \to \infty} P(A_k)$. I suck at proofs so I am ...
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1answer
41 views

f in L^1 implies lim f(x+n) = 0 for a.e. x

Let $f \in L^1(\mathbb{R})$. Prove that for almost every $x \in [0,1]$, the sequence $\{f(x+n)\}_{n=1}^{\infty}$ converges to $0$. I am trying to use the fact that $f(x+n)= \frac{d}{dx}\int_{a}^{x+n} ...
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3answers
174 views

If $\int_A f\,dm = 0$ for all $A$ having some fixed measure $C$, then $f = 0$ almost everywhere

Let $ f \in L^1[0,1]$. Assume that there is a constant C, with $0 < C < 1$, such that for every measurable set $A \subset [0,1] $ with $m(A)=C$, we have $ \int_{A} f dm = 0 $. Prove that $...
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89 views

Problems with Proof of Jensen's Inequality (Durrett's “Probability Theory and Examples”)

I have some questions concerning the proof of the Jensen's Inequality I found in Durrett's "Probability Theory and Examples" [pp.23-24]. In the following there is the proof, with the questions I have ...
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1answer
42 views

Is this function Lebesgue-integrable?

The following function: $$f(x,y)=\dfrac{x^2-y^2}{(x^2+y^2)^2}$$ Lebesgue-integrable in $\Omega:=(0,1)\times(0,1)$? My approach is to convert in polar coordinates and I don't know what would be the ...
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0answers
28 views

Lebesgue measure vs Lebesgue-Stieltjes measure

Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $\mathbb{R}$? Thank you....
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1answer
53 views

Limit of an integral that resembles the Riemann-Lebesgue Lemma

Calculate the following limit $$\lim_{n\to\infty}\int_{0}^{\infty} \frac{x^{n-2}}{1+ x^{n}}\cos(n\pi x) dx$$ I tried by broke down the integration from 0 to 1 and 1 to infinity. I am done with 0 to ...