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

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2
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97 views

Open sets in the topology of weak convergence

I do have various questions regarding the topic of probability measures on polish spaces in general, thus I am trying to divide them in “small” subquestions. Hence, this is my first question on this ...
2
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0answers
64 views

Law of large numbers for a continuum of random variables

Consider a continuum of random variables such that each takes the value $1$ with probability $p$ and $0$ with probability $1-p$. The random variables should be essentially pairwise independent. Sun ...
2
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1answer
56 views

a.e.-defined integrable functions on $X$.

I'm reading the book Folland-Real Analysis. On page 54 after the proposition 2.23 it is written: With this in mind, we shall find it more convenient to redefine $L^{1}(\mu)$ to be the set of ...
2
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0answers
35 views

can the emphasis on “smallest” in the monotone class theorem be ignored in applications?

The monotone class theorem states that for any algebra of sets $\cal A$ one can construct the smallest monotone class generated by this class ${\cal M}(\cal A)$. This smallest monotone class is also ...
2
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0answers
24 views

Borel functions and continuous functions [duplicate]

Suppose we have a set $A\subset\mathbb {R}$ and let $f\in\mathcal{B}(A)$ and $g\in\mathcal{B}_b(A)$ (Borel function on $A$ and bounded Borel function on $A$, resp.) Is it possible to approximate $f$ ...
2
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0answers
47 views

Show that following mapping is measurable

Let $\{X_{n}\}_{n\geq 1}$ be a sequence of random variables on a probability space, $(\Omega, \mathcal{A},\mathbb{P})$. Define the following mapping: $X : \Omega \rightarrow \mathbb{R}^{\infty}$ ...
2
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65 views

Necessary and sufficient conditions for convergence in $\mathcal{L}^1$ (given convergence in measure)

Is the following "conjecture" true? (I'm pretty sure my proof for this is sound and so i'd like to be assured that it's not wrong at least). Let $(X, \mu)$ be a measure space and $f_n, f \in ...
2
votes
2answers
61 views

If $f$ is measurable and $\int f < \infty$, then $f(x) < \infty$ a.e.

I am looking for a hint for what should be a simple proof, but once again I am missing the key connection. Please don't provide a complete solution, nudge me to discover what I am missing. If $f$ ...
2
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1answer
58 views

if a function sequence converges a.e, then it uniformly converges a.e

Assume $f,f_1,...,f_n,... :X\to\mathbb C$ are measureable functions, $f_j\overset{a.e}{\to}f$, and $X$ is $\sigma-$finite. Then there are $\{E_k\}_{k=1}^\infty$ such that $f_j\to f$ uniformly on ...
2
votes
0answers
30 views

Differentiating w.r.t. the boundary of the expected value

I need to solve for general and cont. diff. pdf $g(x)$ $$\frac{d}{db} \int_0^b xg(x)dx$$ Standard Leibnitz rule would give me $$b g(b) + \int_0^b 0 dx$$ the result makes sense - but I'm not sure ...
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44 views

If $A_i$ is measurable, prove that $B=\bigcap_{i=1}^\infty \bigcup_{n=i}^\infty A_n$ is a measurable set

Question If $A_i$ is measurable, prove that $$B=\bigcap_{i=1}^\infty \bigcup_{n=i}^\infty A_n \quad C=\bigcup_{i=1}^\infty \bigcap_{n=i}^\infty A_n$$ is a measurable set. Attempt Since a ...
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0answers
88 views

The difference between Riemann and Lebesgue measures [closed]

What is the difference between the Riemann and Lebesgue integrals ? I'm quite confusing between these two.
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0answers
42 views

$||f_n-f||_1 \to 0 $ iff $ \int| f_n|\ to \int|f| $ if $f_n \to f $ a.e. [duplicate]

Assume $f_n , f \in L^1 $ and almost every where we have $ f_n \to f$ then I want to show that $\int|f_n-f| \to 0$ iff $\int|f_n| \to \int|f|$ One side is abvious by trinagle inequality , for the ...
2
votes
1answer
55 views

Questions about shift-invariant measures in ${\bf N}$

Let $P$ be a shift invariant diffused probability measure defined on powerset of all natural numbers ${\bf N}$(see, van Douwen, Eric K. (1992). Finitely additive measures on ${\bf N}$. Topology ...
2
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0answers
59 views

Unique Ergodicity

Show that unique ergodicity is a topological invariant. Is arguing as follows an overkill (hopefully if the logic is correct --- I have a feeling that there has to be a way a T-invariant measure has ...
2
votes
2answers
213 views

Disjoint system of sets with positive measure is countable

the question is: Let A be a given collection of disjoint measurable subsets of R^d , all of which have positive measure. Show that A is countable. so I was able to prove that each of those subsets ...
2
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0answers
38 views

Ono-to-one-function vs zero-determinant

Let $G : \mathcal{S} \rightarrow \mathcal{T}$ be a continuous, differentiable and bijective function from a manifolds $\mathcal{S}$ to a manifolds $\mathcal{T}$. Let $J(G(x))$ be the Jacobian of $G$ ...
2
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0answers
46 views

Does the Orlicz Norm always make the corresponding integral 1?

Let $\Psi: [0,\infty] \to [0,\infty]$ so that $\Psi$ is convex, and strictly increasing with $\Psi(0) = 0$ and $\Psi(\infty) = \infty.$ If $(X,A,\mu)$ is a measure space, then we define ...
2
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0answers
32 views

Is $\sigma(X_1X_2)\subseteq\sigma(X_1,X_2)$ true?

Given two real valued random variables $X_1$ and $X_2$, I think that $\sigma(X_1X_2)\subseteq\sigma(X_1,X_2)$, but I can't prove it. Here $\sigma(X)$ denotes the $\sigma$-algebra generated by the ...
2
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0answers
74 views

Continuity of Cantor-Lebesgue Function: How to conclude?

Question: Show that the Cantor-Lebesgue function is continuous for $x_0 \in \mathcal{C} - \{0,1\}$, where $\mathcal{C}$ denotes the Cantor set. (The Cantor-Lebesgue function $\varphi$ is an ...
2
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0answers
50 views

A question about product $\sigma$-algebra

Assume $(X,\mathscr{G})$ is a measurable space where $X$ is a finite set and $\mathscr{G}$ is just the power set. Let $\Omega$ be the infinite product of $X$ and $\mathscr{F}$ be the usual product ...
2
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0answers
40 views

Upper limit proof that I don't understand

I don't understand the part where they mention $w \in$ lim $A_n$ I don't understand the part of the fact that w is in finitely many of the $A_n$. I understand that upper limit is union of ...
2
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0answers
32 views

Proving a sum is finite using Equidistribution

Let $\phi:\mathbb{R\to R}$, be an integrable function with finite integral on $[0,1]$($\int_{[0,1]}\phi(x)dm<\infty$) and $\phi(x)=\phi(x+1)\forall x\in \mathbb{R}$. Prove that ...
2
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0answers
30 views

If $E$ is a null set, then $E$ can be covered by a sequence of intervals $(I_n)$ such that $\sum\limits_{n=1}^{\infty} m^{*}( I_n)<\infty$. [duplicate]

The following is a problem from Real analysis by N. L. Carothers. 16.19. For $E\subset [a,b]$, show that $m^{*}(E) = 0$ if and only if $E$ can be covered by a sequence of intervals $(I_n)$ such ...
2
votes
1answer
104 views

Product of two uniformly square integrable random variables

Take two classes of uniformly square integrable random variables $\lbrace X_t:t\in T\rbrace$ and $\lbrace Y_t:t\in T\rbrace$. Is the class $\lbrace X_tY_t :t\in T\rbrace$ uniformly integrable? My ...
2
votes
0answers
48 views

Natural measure of a chaotic system and its prime orbits

Evidently, "the natural measure associated with a chaotic attractor gives the fraction of the time that the long orbit on the attractor spends in any given region of state space." An illustrative way ...
2
votes
0answers
147 views

Essential supremum definition

http://en.wikipedia.org/wiki/Essential_supremum_and_essential_infimum From the definition here I'm unable to tell, whether we have to take the infimum, or whether we could take a minimum - does a ...
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0answers
147 views

Convergence of probability measures on the space of real Radon measures

Consider the space $C_c := C_c(\mathbb{R})$ of compactly supported continuous functions with the inductive limit topology and $C_c'$ its topological dual which can be identified with the space of real ...
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25 views

Equivalence of sigma algebras in averaging process

In a proof I came across the following note and I don't know how it can be shown: Edit: Let $\mathcal{S} = \{x_1,\dotsc,x_N\}$ be a finite set of $N$ real points (not necessarily distinct). ...
2
votes
1answer
31 views

Old qualifier problem clarification (Probability related)

I'm trying to make sense of what is being asked in the question. What does the set $E_{n}^{\epsilon}$ represent? Let $\mathbb{P}$ be a probability measure on $\mathcal{B}(\mathbb{R})$. Let ...
2
votes
1answer
82 views

Is there any $\sigma$-algebra where its elements are equal to a finite disjoint union of generators?

Let $X$ be a set and $\mathcal{B}$ be a family of subsets of $X$. Let $\Sigma$ be the smallest $\sigma$-algebra that contains all elements of $\mathcal{B}.$ Under which assumptions it holds that for ...
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43 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
2
votes
0answers
34 views

Computing $\pi_1(\text{Pr}(S),\mathbb{P}_0)$

Let $(S,d)$ be a complete separable metric space, and consider the space $\text{Pr}(S)$ of probability measures on $S$ that are defined on Borel sets arising from the metric $d$. Now endow ...
2
votes
1answer
117 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[\lim_{n \to ...
2
votes
1answer
41 views

Complex Measures: Absolute Continuity [closed]

Note: This is a lemma for: Spectral Measures: Riemann-Lebesgue Given a positive measure: $$\lambda:\mathcal{A}\to[0,\infty]$$ Consider a complex measure: $$\mu:\mathcal{A}\to\mathbb{C}$$ How to ...
2
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0answers
33 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
2
votes
2answers
228 views

Measure Theory Book for My Background / Need

My current Math background is as follows: 1) Read first 7 chapters of Rudin "Principles of Mathematical Analysis" and solved a lot of the given problems. 2) Completed Munkres "Analysis on Manifolds" ...
2
votes
1answer
58 views

image or projection of a universally measurable set

Suppose $A \subset [0,1]\times[0,1]$ is universally measurable. Is it true that its projection to the first coordinate is a universally measurable subset of $[0,1]$? What is known is that the ...
2
votes
1answer
294 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
2
votes
0answers
15 views

Measurability preserving operators on $L^2$

Given $L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$, a $\sigma$-algebra $\mathcal{G} \leq \mathcal{F}$, a function $f \in L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$ which is $\mathcal{G}$-measurable, ...
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votes
0answers
135 views

Completeness of $ L^{p} $ spaces and “rapidly Cauchy” sequences

http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf In the book of Royden, the completeness of $ L^{p} $ spaces has been done using what he calls "rapidly ...
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votes
0answers
60 views

Show that $d+1$-dimensional Lebesgue measure of set $G$ equals $0$

Let $D \subset \mathbb{R}^d$ and let $f:D \rightarrow \mathbb{R} $ be measurable function. Let $G=\{(x_1,x_2,\ldots,x_d,f(x_1,x_2,\ldots,x_d))\in \mathbb{R}^{d+1}:(x_1,x_2,\ldots,x_d)\in D \} $ be the ...
2
votes
0answers
58 views

Riemann and Lebesgue improper integral Proof

I've been trying to find some notes on the following statement: Let $f:(a,b] \to \mathbb{R}$, $f\geq 0$, and $f\in\mathcal{R}[a+\epsilon , b]$ for any $\epsilon>0$. Then $\int_a^bf=\lim_{\epsilon ...
2
votes
0answers
33 views

Is $\int_{E_k}f\underset{k\to\infty }{\longrightarrow }\int_E f$

If $E_k$ is mesurable such that $E_k\to E$ with $E$ measurable and if $f$ integrable can I do (by theorem convergence dominated) $$\int_{E_k}f\underset{k\to\infty }{\longrightarrow }\int_E f\ \ ?$$ I ...
2
votes
1answer
51 views

If $f$ is integrable on $\mathbb R$, there exists a nullset $N$ with $\liminf_{n\to\infty}f(x+n)=0 \ \text{ for all } x\in\mathbb{R}\setminus N$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lebesgue integrable. Show that there exists a nullset $N$, such that $$\liminf_{n\to\infty}f(x+n)=0 \ \text{ for all } x\in\mathbb{R}\setminus N$$ ...
2
votes
0answers
158 views

Question 39 in Folland's Real Analysis chapter 3

The question "If {$F_j$} is a sequence of nonnegative increasing functions on $[a,b]$ such that $F(x)= \sum_1^\infty F_j(x) < \infty$" for all $x \in [a,b]$, then $F\prime(x)=\sum_1^\infty ...
2
votes
1answer
144 views

Correspondence between countably generated sigma algebras and partitions

Let X be a standard Borel space and $\mathcal C, \mathcal D$ be countably generated sub sigma algebras of the Borel sigma algebra of X. Suppose that for each $x \in X$ we have $[x]_{\mathcal C} ...
2
votes
0answers
85 views

Regularity energy minimizing harmonic maps

I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38). This chapter is dealing with the theory ...
2
votes
0answers
213 views

Measure Theory - Folland - Problem 2.7

This is a problem from Real Analysis - Modern Techniques and Their Applications, by Folland. I'm trying my best here, but it's hard to solve. PS: The measurable space is $(X,\mathcal{M})$. To me, ...
2
votes
1answer
37 views

Converging exponentially in probability implies convergence with probability one?

When I read the paper On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimate, the theorem 1 in it states something like If for every $\epsilon > 0$ there exists ...