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

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2
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1answer
20 views

Motivation for Definition of Measurable Function

I'm having trouble understanding why a function is defined as "measurable" if the preimage of every measurable set is measurable. I see the parallel to the definition of continuity, and the latter ...
2
votes
1answer
16 views

Subset of $A$ of arbirary small measure

Let $\mu(A)>0$. Show that for arbitrary small $\epsilon>0$, there exists a subset $B$ of $A$ such that \begin{align} 0 < \mu(B) < \epsilon \end{align} Assume that $\mu$ is not an attomic ...
0
votes
0answers
38 views

$\sigma$-additivity of an abstract measure

I know and have been able to prove the following lemma: Let $X$ be a set and $\mathfrak{M}$ a $\delta$-ring of subsets of $X$. The set $A\subset X$ is defined as measurable with respect to ...
0
votes
1answer
22 views

Show that $\sigma(S(C))=\sigma(C)$.

Let $ S(C)$ be an algebra generated by $C$ and let $\sigma(S(C))$ be a sigma algebra generated by $ S(C)$. Show that $\sigma(S(C))=\sigma(C)$. I can show $\sigma(C) \subset \sigma(S(C))$ this is ...
0
votes
1answer
42 views

Uncountable and Unbounded set of measure 0?

In my Real Anaysis course, the instructor posed a question. He asked to either give an example or to explain why it is not possible. He asked about an uncountable and unbounded set of measure 0. My ...
2
votes
2answers
32 views

$\sigma$-algebra generate by uncountable collection of subsets.

here is my question: Let $\sigma(F)$ be a $\sigma$-algebra generated by $F$ where $F$ is uncountable collection of subsets of $\Omega$. Let $A \in\sigma(F)$ then there exists a countable ...
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vote
0answers
27 views

Extending a pre-measure on an sigma-algebra?

Consider on $\Bbb{R}$ the family $\Sigma $ of all Borel sets which are symmetric w.r.t. the origin, which is a $\sigma $-algebra. Is it possible to extend a pre-measure $\mu $ on $\Sigma $ to a ...
5
votes
2answers
33 views

Consider on $\Bbb{R}$ the family $\Sigma $ of all Borel sets which are symmetric w.r.t. the origin. Show that $\Sigma $ is a $\sigma $-algebra.

Consider on $\Bbb{R}$ the family $\Sigma $ of all Borel sets which are symmetric w.r.t. the origin. Show that $\Sigma $ is a $\sigma $-algebra. Context: Preparing for my exam Effort: To ...
4
votes
1answer
166 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
2
votes
1answer
47 views

Half open intervals in a Borel $\sigma$-algebra [duplicate]

I am working on the exercise: prove that a right continuous function $\mathbb{R} \to \mathbb{R}$ is Borel measurable. I found that for every $x \in f^{-1}((a,b))$ we must have $$ [x, x + \delta_x) ...
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votes
0answers
21 views

Show that the Radon-Nikodym fails when omitting the assumption of $\sigma $-finitenes.

Let $\lambda $ be the counting measure on the $\sigma $-algebra of lebesgue measurable set in $(0,1) $ [Then $\lambda $ is not $\sigma $-finite. ], and let $\mu $ be the lebesgue measure on $(0,1) $. ...
8
votes
1answer
98 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
0
votes
1answer
46 views

Real Analysis - Lebesgue integrable functions

Let $E$ be a measurable set. Suppose $f \geq 0$ and let $E_k=\{x \in E_k|f(x) \in (2^k, 2^{k+1}] \} $ for any integer $k$. If $f$ is finite almost everywhere, then $\bigcup E_k = \{x \in E |f(x)>0 ...
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0answers
20 views

Reference: proof of Cramer-Rao

I'm looking for a detailed reference of dealing with the proof of the multivariate case of Cramer-Rao lower bound.
1
vote
1answer
19 views

For X, Y bounded random variables, $E[X^m Y^n] = E[X^m]E[Y^n]$ for all $m, n \geq 0$ implies X, Y are independent

Assume that X, Y are two bounded random variables. If for any integers $m, n \geq 0$, $E[X^m Y^n] = E[X^m]E[Y^n]$, then X and Y are independent. I've worked out that, for sure, if $E[f(X)g(Y)] = ...
0
votes
0answers
9 views

What does it mean for a stochastic process to be measurable?

In my first class of Stochastic calculus the professor said that a process X is measurable if the map $(t,\omega) \mapsto X_t(\omega)$ is measurable from $(\mathbb{R^+ \times ...
1
vote
2answers
23 views

Proving the bound $P(|X| \geq x) \leq (M_X(c) + M_X(-c)) e^{-cx}$ under certain conditions

If the moment generating function of a random variable X, $M_X (\lambda) = E[e^{\lambda X}]$, is defined for $|\lambda| < \delta$ with some $\delta > 0$, then $P(|X| \geq x) \leq (M_X(c) + ...
0
votes
1answer
18 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
2
votes
0answers
17 views

Set of zeros of derivate - Lebesgue measure [duplicate]

I'm currently struggeling with the following: Let $\lambda$ be the Lebesgue measure and $f \in C^2[0, 1]$. Show: If $\lambda(\{x \in [0, 1]; f(x) = 0 \}) > 0$, then $\lambda(\{ x \in [0, 1]; f'(x) ...
0
votes
1answer
44 views

$P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$ for all $a < b$ implies that $lim_{n \rightarrow \infty} X_n$ exists a.e.

Suppose for $a<b$ we have $P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$. Then $lim_{n \rightarrow \infty} X_n$ exists a.e. but may be infinite. Here "i.o." means "infinitely often"; for any ...
0
votes
1answer
20 views

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 ...
3
votes
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..
2
votes
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 ...
0
votes
1answer
25 views

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 ...
5
votes
1answer
48 views

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 $$\sum_{n=1}^{\infty}\frac{1}{2^n}X_n = \mu \; a.s$$ ...
2
votes
1answer
31 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 ...
1
vote
1answer
11 views

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? ...
1
vote
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 ...
2
votes
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 ...
1
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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 $\gamma : \mathcal{B}(\Bbb{R}) \rightarrow \{0,1\}$ ...
0
votes
1answer
22 views

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 ...
3
votes
0answers
41 views

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 ...
0
votes
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
17 views

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 ...
2
votes
1answer
42 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 ...
2
votes
1answer
45 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 ...
0
votes
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
39 views

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
28 views

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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votes
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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votes
0answers
17 views

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 ...
0
votes
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 ...
1
vote
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
votes
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 ...
4
votes
2answers
40 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
votes
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
votes
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 ...
0
votes
0answers
37 views

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 ...
2
votes
1answer
21 views

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 ...