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

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Essential Uniform Convergence Implication

Greetings Mathematics Community. I believe that I am thinking too hard about the following problem and would like some guidance in solving it. Let $X$ have finite measure and let $f_n:X \to ...
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3answers
73 views

$m\{x\in [0,1]:f'(x)=0\}>0$ [closed]

Let $f\in C^2\{[0,1],\mathbb{R})$ and $m\{x\in [0,1]:f(x)=0\}>0$. Prove that $$m\{x\in [0,1]:f'(x)=0\}>0,$$ where $m$ denotes Lebesgue measure. I don't have any clue to solve this exercise..
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2answers
36 views

A measure is sigma-finite if, and only if, there exists a integrable function w such that its image is contained in (0,1)

I have to prove the following proposition: Consider a measure space $(S,\Sigma,\mu)$. Prove that $\mu$ is $\sigma$-finite if, and only if, there exists $w\in\mathcal{L}^1(S,\Sigma,\mu)$ such that ...
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1answer
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Example of semi-algebras that are not algebras

I know that every algebra is as semi-algebra, and the book (A course in Real Analysis, McDonald and Weiss) tells me that the opposite is not true: not every semi-algebra is an algebra. Why not? A ...
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1answer
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Prove that this sequence converges almost surely

Suppose that $(X_n)_{n\ge1}$ is a sequence of independent random variables with $E[|X_n|] < \infty$ for all $n$ and $E(X_n) = \mu$. Prove that $$\sum_{n=1}^{\infty}\frac{1}{2^n}X_n = \mu \; a.s$$ ...
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1answer
30 views

Subset and optional times

The below is a well known fact but can anyone help me to prove it? If I have a right continuous filtration and $\eta$ is an optional time, how can I show that if $\eta\leq t$ then $\mathcal{F}_\eta ...
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1answer
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Modification of Set Function in Construction of Lebesgue Measure

Suppose in the construction of Lebesgue measure we replace the set function $\mu((a,b))=b-a$ with $\mu((a,b))=\sqrt{b-a}$. What can we say about $\mu^*$ and the $\sigma$-algebra of measurable sets? ...
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1answer
35 views

Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
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0answers
34 views

Subadditivity of Lebesgue-Stieltjes measure

Kolmogorov-Fomin's Элементы теории функций и функционального анализа define an elementary set as the finite union of intervals of the form $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$, $[a,a]:=\{a\}$ and ...
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1answer
52 views

Is this a bug in the solution manual of Measures, Integrals and Martingales by Rene Schilling?

I am reading the solution of problem 4.3 of Measures, Integrals and Martingales by Rene Schilling. Problem 4.3. Show that the function $\gamma : \mathcal{B}(\Bbb{R}) \rightarrow \{0,1\}$ ...
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1answer
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A property of Radon-Nikodym derivatives

Is the Radon-Nikodym derivative linear in countably infinitely many input measures? That is, if fi is a Radon-Nikodym derivative of the measure vi w.r.t. u for i from 1 to infinity, then is it true ...
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0answers
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Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
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2answers
35 views

Suppose $B \in \mathcal B(\mathbb R)$. Why is it neccesarily true that $-B := \{-x \mid x \in B\} \in \mathcal B(\mathbb R)$?

Let $\langle\mathbb R, \mathcal B(\mathbb R)\rangle$ be the $\sigma-$algebra of Borel sets in $\mathbb R$. Suppose $B \in \mathcal B(\mathbb R)$. Why is it necesarily true that $$ -B := \{-x : x ...
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0answers
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Is sigma-finiteness required?

Suppose that f and g are two extended-real valued measurable functions on an arbitrary measure space, such that the integral of g dominates that of f on every set A of the sigma-field. Is it true ...
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1answer
41 views

Exact statement of the Radon-Nikodym Theorem

I am a bit confused about the exact statement of the Radon-Nikodym Theorem. Suppose that in the usual setup, $v \ll u$. Does it require both $v$ and $u$ to be sigma-finite, or only $u$ to be sigma ...
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1answer
43 views

Construct a Borel set on R such that it intersect every open interval with non-zero non-“full” measure

This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis. The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and ...
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1answer
28 views

Probability and Measure: Sigma-finite

What is a example that shows that $\mu$ $\sigma$ -finite does not imply $\mu \cdot T^{-1}$. I have a basic understanding of what $\sigma$-finite means but if $\mu$ is finite implies $\mu \cdot T^{-1}$ ...
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0answers
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Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory

I'm taking stochastic probability class but I'm now only taking analysis (with Rudin's PMA) class. The stochastic probability class doesn't depend heavily on the theoretic structures: rather, the ...
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1answer
13 views

Finding a bound for the maximum function

the following problem says: Show that if is f an integrable function in $\mathbb{R}^d$ and not identically null, then $$f^*(x)\geq\frac{c}{|x|^d}$$ where $c>0$, $|x|\geq 1$ and $f^*(x)=\sup_{x\in ...
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0answers
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Reciprocal of measurable function is measurable

Let $f(x)$ be a measurable function and define $$g(x)= \begin{cases} \frac{1}{f(x)}, & f(x) \not= 0 \\ 0, & f(x)=0 \\ \end{cases} $$ Show that $g(x)$ is also measurable. Here's my reasoning ...
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3answers
46 views

Complex Lebesgue integral, property

Lets say that you for real functions have proved that: $|\int_{\Omega}fd\mu|\le \int_{\Omega}|f|d\mu$. How do I then prove that it also holds for complex-valued functions? I guess this amounts to ...
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Trouble with minor detail in proof

We proved in class the countable subadditivity of a general measure. My question is at the end. Statement: If $\{A_k\}_{k=1}^{\infty} \subseteq \mathscr{F}$ and $\cup_{k=1}^{\infty}A_k \in ...
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1answer
31 views

a dominated function for integral

I have a question :I wil show from dominated convergence theorem that equality : $\int^\infty_1\frac {log(x)}{x(x-1)} dx =\sum_{k=1}^{\infty}\frac{1}{k^2}$ I know that ...
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1answer
23 views

Show that $P(X)$ is a sigma algebra.

Show that $P(X)$ is a sigma algebra. First of all, surely this makes totally sense. But I'm wondering how you would actually prove this. How do you prove something as trivial that $P(X)$ is ...
2
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1answer
34 views

A basic question from measure theory

I'm very new in measure theory and I have the following question: Let $(X, \Sigma, \mu$) be a probability measure, that is, $\mu (X)=1$. Two measurable, real valued functions $f$ and $g$ on $X$ are ...
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2answers
39 views

Cauchy convergence in probability implies the existence of a (finite a.e.) limit X

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an X (finite a.e.), such that $X_n$ converges to X in probability. The problem's hint suggests ...
2
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0answers
25 views

Outer Measure in Cantor-like Set

Consider the Cantor-like set $C$ resulting from the Cantor-like construction, which starts with $k$ disjoint closed intervals $\delta_i$, $k \ge 2$, $i=1,\dots,k$ of the unit interval. Given an ...
3
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1answer
34 views

Do positive integrals imply positive function in this case?

Suppose that $f: \mathbb{R} \to [0, \infty)$ is Borel measurable and satisfies $\|f\|_\infty \le 1$ and $\|f\|_1 = 1$. If $$\int_a^b \! f(x) \, dx > 0$$ for all $a < b$, does it necessarily ...
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Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
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1answer
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series of the integrals converges then the series converges almost surely

I know this was asked but I want a proof of this without using Fubini theorem. Anyway the first part of the problem can't be concluded using Fubini. I don't know how to do it :/ Let $f_k:\mathbb R ...
3
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1answer
46 views

What is meant by“…on se ramène par régularisation…”?

I am currently attempting to translate the paper 'Sur l'équation de convolution $\mu = \mu \ast \sigma$' by Choquet and Deny. In the paper, a locally compact abelian group $G$ and a positive measure ...
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0answers
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Not sure if I understand the significance of support in these theorems.

I am just beginning to study the Lebesgue integral, and our building our way up to it. Right now we are defining the integral for bounded functions supported on a set of finite measure. In the ...
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0answers
28 views

Measure theory mapping sets to groups?

This is a question from a physicist wondering if a certain idea in mathematics has been developed. Intuitively, suppose I have a number of objects distributed in space. I want a function that given a ...
1
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1answer
35 views

$\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$ implies that random variables $X_n$ are uniformly integrable

$X_n$ are uniformly integrable if $\lim_{R \rightarrow \infty} \sup_n E[|X_n|,|X_n| \geq R] = 0$. Show that if $\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$, then $X_n$ are uniformly integrable. ...
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Total variation of a complex measure [duplicate]

Let μ be a complex measure on a measurable space (X,Σ) . Let |μ| be the total variation of μ , defined by |μ|(E)=sup{∑j |μ(E j )|:{E j } is a countable pairwise disjoint, Σ-measurable partition of ...
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0answers
26 views

Almost sure downward convergence with some conditions implies convergence in $L^1$

If $X_n\downarrow X$ a.s., each $X_n$ is integrable and $inf_n E[X_n] > -\infty$, then $X_n \rightarrow X$ in $L^1$. As far as I know, "$X_n\downarrow X$ a.s." means that for every n, $X_n ...
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1answer
32 views

Two stochastic processes with the same distribution inducing different measures

I am currently reading Strook's $\textit{Probability Theory: An Analytic View}$, and I am confused by the following statement on page 156: "I take for $D(\mathbb{R}^N)$ the measurable structure given ...
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Show that $\int f d(\lambda + w \mu) =\int f d \lambda + \int fw d \mu $, where $w $ is a bounded function.

I don't understand the following, from Rudins real and complex analysis. Assume $\lambda $ is a positive bounded measure, and associate to the $\sigma-$finite measure $\mu $ a function $0< w(x) ...
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0answers
44 views

$f$ absolutely convergent on $\mathbb{N}$ iff integrable

I am looking for some hints on the following Let $X$ be the set of all positive integers, $\Sigma$ the class of all subsets of $X$, and $\mu(E)$ (for any $E\in \Sigma)$ the number of points in $E$. ...
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1answer
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What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
2
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1answer
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continuous-time version of fatou's lemma

I have just read a textbook on stochastic processes that implicitly uses the fact that \begin{equation} \int \liminf_{t \to \infty} f_t \leq \liminf_{t \to \infty} \int f_t, \end{equation} for ...
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1answer
27 views

Simple function sequence for measurable functions with range $[0,1]$.

Let $[0,1]$ have the usual topology. Consider $f: X \rightarrow [0,1]$ such that $f$ is measurable. Fix $n$. I want to show that there is a measurable simple function $\phi_{n}$ such that ...
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1answer
26 views

Supremum of measurable functions

If $\{f_n\}$ is a sequence of measurable functions on the same measurable set then: i) $\sup_{1\le i\le n} f_i$ is measurable for each $n$. ii) $\sup f_n$ is measurable. I don't quite follow the ...
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2answers
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Sequence of Measurable Functions (Unsigned and Complex-Valued)

I am having several difficulties in solving the following problem about measurable functions: Let $(X,\mathcal{B})$ be a measurable space. If $f_n : X \to [0,+\infty]$ are a sequence of measurable ...
3
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1answer
56 views

Borel measurable functions in measure theory

Suppose that $f$ is a function on $\mathbb{R}\times \mathbb{R^k}$ such that $f(x,\cdot)$ is Borel measurable for each $x\in \mathbb{R}$ and $f(\cdot,y)$ is continuous for each $y\in \mathbb{R^k}$. For ...
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2answers
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What is meant with a $\sigma $-finite measure $\mu $ can be replaced by finite measure in the sense that $d \bar {\mu}=w d \mu $?

I don't understand the following. Before proving the Radon-Nikodym theorm, Rudin proves a lemma that say that $\mu $ is a $\sigma $-finite measure on a $\sigma $-algebra, then there is a function $w ...
3
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1answer
39 views

Convergence in equivalent probability measure

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A_n$ be a sequence of events such that $P(A_n)$ converge to 0. If $Q$ is an equivalent probability measure of $P$, does it mean that $Q(A_n)$ ...
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1answer
121 views

Infinite product of probability measures is a premeasure

This is an exercise from Real Analysis by Stein (Chapter 6, Exercise 15). Given infinitely many measure spaces $X_i$, each of which has measure 1, one can define an algebra on the product space ...
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For a measure $\mu $ *does it exists* a sequence $\{E _n \} $ of sets such that $\mu(E _n)<1/2^n $ for every $n $?

If $\mu $ is a measure defined on a measure space $(X, M, \mu ) $, can I claim that there exists a sequence $\{E _n \} $ of set in $M $ such that for every $n $, $\mu ( E _n) <1/2^n $? If so, what ...
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0answers
14 views

Measurability of $u:[0,T] \to X$ where $u$ is in Bochner space

Let $f \in L^p(0,T;X)$ where $X$ is a separable Banach space. So $f$ is a Bochner function and hence Bochner measurable, meaning that there is a sequence of measurable countably-valued functions that ...