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

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İntersection of Dynkin System

What is the difference between Dynkin system and sigma algebra? I am confused with Dynkin System.. and how I can show that the intersection of Dynkin system is also a Dynkin system. Thank you for ...
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0answers
3 views

Measure theory, Showing a set A is $\mu^*$-measurable

How should I approach the following problem: Let $A\subset X$, show $A$ is $\mu^*$-measurable if there exist $B$ $\mu^*$ measurable such that $\mu^*(A-B)<\epsilon$ for each $\epsilon>0$.
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0answers
12 views

Lebesgue integral and iterated integral

I am learning lebesgue integral at the moment, and come across a question in homework, but find it really confused. The question states: I first tried to compute the iterated integral by Riemann ...
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3answers
25 views

$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle} C)$

Show that $$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle}C)$$ where $\mathrel{\triangle}$ indicates the symmetric difference I cannot write my idea, because ...
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0answers
9 views

upper lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable.Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...
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0answers
15 views

$\mu(A) = \mu(\overline{A})$ For $\mu$ A Radon Measure, $A \in B_b(E)$

Let $\mu$ be a Radon measure, $E$ a polish, locally compact topological room. $$B_b = \{B \in B(E) : B\text{ is relative compact}\}$$ With $B(E)$ being the Borel $\sigma$-algebra. Is it true that ...
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1answer
24 views

Check/Prove if random variable

I am having some doubts on how to prove/check if something is a random variable. The question I am trying to solve currently is: 1) Consider X a random variable on space (Ω,F,P). If [X] is the ...
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0answers
9 views

Simple functions dense in vector valued space $L^2([0,1]^2,\mu, R^n)$

Suppose $([0,1],\Sigma, \lambda)$ is a probability space and $\mu=\lambda\times \lambda$ is the product measure on $([0,1]^2,\Sigma\times \Sigma)$. $f\in L^2([0,1]^2,\mu, R^4)$ is a vector valued ...
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0answers
10 views

Approximating an element of a product sigma algebra by rectangles independent to a sub sigma algebra

Let $(S,\mathcal{S},\mu)$ be a finite measure space and $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Denote the product space $(S\times \Omega, \mathcal{S} \times \mathcal{F},\mu \times ...
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1answer
17 views

Lebesgue integral of absolute value of sequence of functions

I am working on a problem$^{(*)}$ on Lebesgue integral looks like this: Given that both $f_n$ and $f$ are integrable, $f_n \longrightarrow f$ a.e., and $\int|f_n| \longrightarrow \int |f|$. Show ...
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0answers
31 views

using Borel-Cantelli Lemma to show that $\lim_{n\to \infty}P(A_n)$ exists and $\lim_{n\to \infty}P(A_n) =P(\lim\sup A_n)$

**Question: ** if $$\sum_{N=1}^\infty P(A_nA_{n+1}^c )\lt \infty$$ use Borel-Cantelli Lemma to show that a) $\lim_{n\to \infty}P(A_n)$ exists b) and $\lim_{n\to \infty}P(A_n) =P(\lim\sup A_n)$ ...
2
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2answers
43 views

how that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$

Question: Let $\{A_n\}$ be a sequence of independent events in a probability space $(\Omega, F, P)$ show that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$ I tried solving this ...
7
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0answers
57 views

Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second ...
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2answers
26 views

Why does $\left\{\left|X-\sum_{i=1}^mX_i\right|\le\delta\right\}$ depend only on $X_i,i\ge m+1$, if all $X_i$ are independent?

Assumptions Let $(X_i)_{i\in\mathbb{N}}$ a sequence of independent real-valued random variables and $$B_{m,n}:=\left\{\left|X^{(i)}-X^{(n)}\right|\le2\delta \text{ for all }i\in[n,m-1]\text{ and ...
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0answers
44 views

About the differential notation in measure theory

Is there any good reason for which integrating according to a measure includes a $\mathrm d$ as in $\int f\mathrm d\mu$ ? Or is it just a manner to keep formal consistency with the traditional ...
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1answer
14 views

property of borel measures

I am reading chapter 1.3 in weak convergence and empirical processes of Van der Vaart and Wellner. Let $(\mathbb{D}, d)$ a metric space and let $L$ a Borel probability measure. In the proof of the ...
2
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2answers
42 views

Measurable set and continuous function exercise

Let $E \subset \mathbb R^n$ be a measurable set. Show that $f:\mathbb R_{\geq 0} \to \mathbb R$, given by $f(r)=|E \cap B(0,r)|$ is continuous. So, given a fixed $r_0 \in \mathbb R_{\geq 0}$ and ...
4
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3answers
116 views

Graph of real continuous function has measure zero

Let $f\colon[a,b] \to \mathbb R$ be a continuous function. Show that its graph has measure zero. I've tried with the following idea but I got stuck: Let $\epsilon >0$, since $f$ is uniformly ...
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1answer
19 views

Basic problem about measurable sets

Problem Let $E \subset \mathbb R^n$ with $E$ a measurable set, $E=A \cup B$, where $ |B|=0$. Show that $A$ is measurable. Here is what I could do (btw, $ |.|_e$ stands for outer measure) We have ...
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0answers
35 views

Measure theory - circle with perimeter 4 (yes really) [on hold]

I am writing an article about the Lebesgue measure for my wiki and I've talked about the following: $\mathcal{J}^n=\{$of all half-open-half-closed-rectangles in $\mathbb{R}^n\}$ ...
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0answers
45 views

Construction Of A Cantor-like set with certain property [duplicate]

Is there a way to construct a measurable set $E \subset [0,1]$ with the property that for every interval $[a,b] \subset [0,1]$, both $[a,b]\cap E $ and $[a,b]$\ $E$ have positive Lebesgue measure? I ...
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1answer
16 views

Examples of an unbounded measurable subset of finite measure of the p-adic number field

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. A subset of $\mathbb{Q}_p$ is called bounded if it is contained in a compact subset. Let $\mu$ be the Haar measure ...
2
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1answer
50 views

Show that $\mu( \bigcap_{i=1}^n A_i )\gt 0$

let $(\Omega, F, P)$ be a probability space and suppose the sets $A_1, A_2,...,A_n \in F$satisfy the inequality $$\sum_{i=1}^n\mu(A_i) \gt n-1$$ Show that $$\mu( \bigcap_{i=1}^n A_i ) \gt 0$$ I ...
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1answer
22 views

How can a set be a subset of a $\sigma$-algebra

Take the measure space $(X,\mathcal A)$, and a set $S\subset X$. What does $S\subset \mathcal A$ actually mean? $A\subset B$ if every element of $A$ is also in $B$. But elements of $\mathcal A$ are ...
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0answers
13 views

Measurability of vector valued functions and the hyper plane separation counterpart

Let $([0,1],\Sigma,\mu)$ be a probability space where $P$ is $\sigma$-finite. Define a product measure $\lambda=\mu\times \mu$ on $([0,1]^2, \Sigma\times \Sigma)$ . Consider vector valued functions ...
4
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0answers
33 views

Borel sigma algebra not containing all subsets of $\mathbb{R}$?

Consider the smallest sigma algebra $\mathscr{B}$ generated by all open subsets of $\mathbb{R}$. One would expect that $\mathscr{B}$ contains all subsets of $\mathbb{R}$, but as it turns out, if we ...
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1answer
14 views

If $\mu$ is a (sub-)probability measure on $\mathbb{R}$, then $\mu(\left\{x\right\})=0$ for all continuity points $x$ of the DF of $\mu$

Let $\mu$ be a (sub-)probability measure on $\left(\mathbb{R},\mathcal{B}(\mathbb{R})\right)$ and $F$ be the distribution function of $\mu$. How can we deduce ...
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An assertion in the proof of Riesz Representation Theorem: the Radon measure is zero over $f^{-1}(t_i)$

The Riesz Representation Theorem is this version: $\quad$Let $L:C_c(R^n,R^m)\rightarrow R$ be a linear operator, where $C_c(R^n,R^m)$ denotes the space of continuous map from $R^n$ to $R^m$ with ...
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0answers
42 views

construction of a Lipschitz function [on hold]

I have to solve this exercise.\ Let $A \subset \mathbb {R}$ a lebesgue measurable set with positive lebesgue measure. Find a Lipschitz function $f:\mathbb {R} \to \mathbb {R}$ such that $f(A)$ is a ...
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0answers
33 views

Lipschitz real function, Lebesgue-measurable sets into intervals with positive measure

Just an exercise from a university course: Construct a Lipschitz real function which sends Lebesgue-measurable sets $A\subset \mathbb{R}$ with positive measure $m^1(A)>0$ into intervals $f(A)$ with ...
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2answers
25 views

Marginal Distribution: Integrate a variable out

Suppose we have given the joint density $f_{(X,Y)}(x,y)$ of two random variables $X, Y$, where $f_{(X,Y)}(x,y)=g(x,y) \mathbb{1}_{y > t}$. Now we want to compute the marginal density of $X$, ...
1
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1answer
19 views

Estimation of the Probability of $\left\{|X^{(i)}-X^{(n)}|\le2\delta\text{ for all }i\in[n,m-1]\text{ and }|X^{(m)}-X^{(n)}|>2\delta\right\}$

Assumptions Let $(X_i)_{i\in\mathbb{N}}$ a sequence of independent real-valued random variables and $$B_{m,n}:=\left\{\left|X^{(i)}-X^{(n)}\right|\le2\delta \text{ for all }i\in[n,m-1]\text{ and ...
0
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0answers
20 views

Define probability measure on the space of multivalued function

Suppose I have a collection of multivalued functions $f:[0,2\pi]\rightarrow\mathbb{R^{3+}}$. It is also known that this space is a vector space. We define distance ...
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1answer
18 views

Clarifying the importance of the quantile function in probability theory

I want to cement my understanding of the quantile function in probability theory and here is the way I understand it. (1) We start off with some probability space $(\mathbb R, B = \sigma(\mathbb ...
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0answers
9 views

Explicit construction of Haar mesure on the p-adic number field

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. Let $\mathcal B$ be the smallest $\sigma$-algebra containing all the open subsets of $\mathbb{Q}_p$. Can we prove ...
5
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2answers
61 views

If $u, u'' \in L^2(0,1)$, is it true that $u' \in L^2(0,1)$?

Let $u \in L^2(0,1)$. If $$u'' \in L^2(0,1)$$ is it true that $$u' \in L^2(0,1)?$$ Why yes/not? If $u, u'' \in L^2(0, 1)$ do not imply that $u' \in L^2(0,1)$, how can I show that $u' \in L^2(0,1)$? ...
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2answers
47 views

If $f:[a,b]\to\mathbb{R}$ is increasing, does it maps Borel sets to measurable sets?

Suppose $f:[a,b]\to\mathbb{R}$ is strictly increasing and left-continuous. Does it follow that $f$ maps Borel subsets of $[a,b]$ to Lebesgue measurable subsets of $\mathbb{R}$? My intuition tells me ...
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1answer
31 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
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28 views

questions about Folland real analysis chapter 1 exercise

Here, E is a Lesbegue-measurable set on the real line. This is the exercise 30, 31 of p. 40 of Folland real analysis. I solved these problems when E is of finite measure, but the problem requires ...
2
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1answer
37 views

Prove increment of Brownian motion is Brownian motion

I am trying to solve the following exercise in Oksendal's book: Let $B_t$ be Brownian motion and fix $t_0\ge 0$. Prove that $$\bar{B_t}:=B_{t_0+t}-B_{t_0};\quad t\ge 0$$ is a Brownian motion. I ...
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0answers
17 views

Solving a problem using of Chebyshev's Inequality

Let $f \in L_{1}(\mu)$ and let $M \gt 0$ such that $$|\frac{1}{\mu(E)}\int_{E}f d\mu| \le M$$ for every $E \in S$ with $0 \lt \mu(E) \lt \infty$. Show that $|f(x)| \lt M$ for a.e $x(\mu)$. Let ...
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1answer
33 views

Finding a new Probability measure

Let $\Omega = \{-2,-1,1,2\}$ and $\mathbb P :\mathscr P(\Omega)\to[0,1]$such that: $$ \mathbb P (F)=\frac{\# F}{\# \Omega}=\frac{\# F}{4} $$ and let $X$ be a random variable such that ...
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0answers
26 views

Measure-theoretic conditional expectation

While working on a homework problem, I am baffled by the following statement: Let $(X,\mathcal{M},\mu)$ be a finite measure space, $\mathcal{N}$ a sub-$\sigma$-algebra of $\mathcal{M}$, and $\nu = ...
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1answer
15 views

meaning of $f_{\chi_{E}}$

given $(X,\mathcal{M})$ a measurable space, I Have $E \subset X$ and $\chi_{E}$ is an indicator function. then what is meant by $f_{\chi_{E}}$ ? I am not very clear with this notation and meaning.
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24 views

For positive measures $\nu_j$, is $\left(\sum_1^\infty \nu_j\right)(E) \le \sum_1^\infty \nu_j(E)$?

I am trying to solve the following homework problem, where the notation $\nu \perp \mu$ means that $\nu$ and $\mu$ are mutually singular: Suppose $\{\nu_j\}$ is a sequence of positive measures. If ...
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1answer
37 views

How can I solve like this exercise in measure theory [on hold]

If $J=\{[a,b[$ : $a \le b$ : $a,b \in R\}$ and $F$ is an continous increasing bounded function on $R$ , and if we put $λ([a,b[)=F(b)-F(a)$ prove that : $$λ(\emptyset) = 0 $$ and if the union of ...
2
votes
1answer
48 views

Can someone clarify something in Fubini's theorem please?

In my notes I have a version of Fubini's theorem which differs from the other forms of it I've seen which seem to all be like the one found on wiki here. Here is the version I have in my notes; Let ...
1
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1answer
21 views

borel measurable functions and measurable functions

Say you are given the Lebesgue measure on the real line and a Lebesgue measurable function $f$. Here Lebesgue measure is a complete measure (defined for some non-Borel set). And note that in the case ...
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0answers
17 views

a question in calculus/real analysis

Can someone help me out to give a bound of the first derivative w.r.t. $x$ of the function $\min_{x,y\in\Omega}\{1,\frac{d(x,\partial\Omega)d(y,\partial\Omega)}{|x-y|^2}\}$?. ...
1
vote
1answer
36 views

prove product sigma algebra contains all open sets

I want to prove that $\mathcal{M} \times \mathcal{M}$ contains all open sets of $R^2$. I know that $\mathcal{M} \times \mathcal{M}$ is the product sigma algebra generated by the measurable rectangles ...