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

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67 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
2
votes
1answer
42 views

Show that is a probability space

Let $ \Omega:= \{(x,y) \in \mathbb{R^2}:0<x,y \leq 1 \}$, let $\mathcal{F}$ be the collection of sets of $\Omega$ such that $$ \mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$$ ...
2
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0answers
55 views

If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
2
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1answer
51 views

Cumulative distribution function implication

How can I prove the following: Let $X$ and $Y$ be two random variables. Suppose that their cumulative distribution functions satisfies $F_X(x)=F_Y(x)$ for all $x$. How can I show that $X$ and $Y$ are ...
2
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0answers
71 views

Extension of an additive function

Let $X$ be a finite set, $S\subset \mathcal P(X)$ such that: $1) X\in S$, $2) A,B\in S, A\cap B=\emptyset \Rightarrow A\sqcup B\in S$ and $3) A,B\in S, A\subset B \Rightarrow B\setminus A \in S$ ...
2
votes
1answer
52 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
2
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32 views

Linear combination of i.i.d random variables

We say that a random variable $X$ satisfies the $(\alpha,\beta)-$condition for some $\alpha>0$ and $\beta>0$ if $$\mathbb{P}(|X|<t)<\alpha t\text{ and }\mathbb{P}(|X|>t)<e^{-\beta ...
2
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116 views

Is “almost all function” a well defined concept?

I am working on a problem which has well defined properties for the vast majority of all PDFs. I would like to make a quantitative statement along the lines of "for almost all distributions, P holds". ...
2
votes
1answer
39 views

Radon measure times a function is still a Radon measure?

Given $\Omega\subset \mathbb R^N$ is open and let function $\varphi$: $\Omega\to [1,+\infty]$, $\varphi\in L^1_{loc}(\Omega)$ be given. Suppose $\mu$ is a finite Radon measure on $\Omega$ and we ...
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48 views

How to write down the probability space of this stochastic process

Consider infinitely repeated coin-toss. Then the probability space can be written as $\Omega=\{H,T\}^\infty$ with its product $\sigma$-algebra. Now let's assume that after each round, there is ...
2
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1answer
92 views

A simple implication of an approximation theorem by Komlós, Major and Tusnády

I have been reading through the PhD thesis of Professor Aue on change point analysis based on invariance principles. There's a particular argument I have not been able to follow: Let ...
2
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0answers
91 views

Question on Egoroff-like theorem

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...
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0answers
59 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
2
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0answers
37 views

Is there a programmatic way to calculate cascaded sigma functions?

Let my format be sigma(function,from,to) = f(n) for example sigma(sigma(1 , j = 1 , j = i) , i = 1 , i = n) = (n^2)/2 ...
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0answers
49 views

How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
2
votes
1answer
88 views

Relating Integration by Substitution to Change of Variables Theorem

I'm having trouble relating the change of variables theorem from measure theory to the integration by substitution formula taught in Calculus. I've always thought they were basically saying the same ...
2
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0answers
28 views

Help with a proof regarding non-measurable set

Please help with the following question where $|E|$ denotes Lebesgue measure. We say set $A⊆\Bbb{R}^n$ is a translation of set $B⊆\Bbb{R}^n$ if $A=B+z$ for some $z∈\Bbb{R}^n$. Let $E$ be a ...
2
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0answers
59 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
2
votes
1answer
47 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
2
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86 views

Computation of integral on parametrized curve

if $U$ and $V$ are two open subset of $\mathbb{R}^{n}$, $\varphi:U\rightarrow V$ a $C^{1}$ diffeomorphism, then we have the change of variable formula for the Lebesgue integral: ...
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0answers
64 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...
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0answers
57 views

If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
2
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1answer
84 views

Is there measurable function defined on unmeasurable set?

In my textbook, Lebesgue measurable function is defined as for every finite $a$, the set $\{x\in E:f(x)>a\}$ is a measurable set of $R^n$. And it further states $E=\{x\in ...
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41 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
2
votes
1answer
138 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [closed]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
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66 views

Approximation by measure with finite support

Can a Borel probability measure on a Polish space be arbitrarily approximated in the total variation metric by a probability measure with finite support?
2
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1answer
48 views

measure-preserving transformations are spectrally isomorphic

If $(X_{1}, \mathcal{B}_{1}, m_{1})$ and $(X_{2}, \mathcal{B}_{2}, m_{2})$ are probability spaces together with measure-preserving transformations $T_{1}:X_{1}\to X_{1}$,$T_{2}:X_{2}\to X_{2}$. How ...
2
votes
1answer
49 views

Relation between two p-norms

While it's a well known that any two norms are equivalent for a finite dimensional normed linear space, I've been trying to derive the bounds for the case $X=\mathbb{R}^n$ and $l_p$-norms. Let $1 ...
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40 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
2
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1answer
66 views

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Problem Statement Let $\{f_n\}$ be a sequence of real-valued, measurable functions on $[0,1]$ that is uniformly bounded. Show that if $A$ is a Borel subset of $[0,1]$ then there exists subsequence ...
2
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0answers
84 views

Inner measure (inner set function) on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
2
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0answers
31 views

Prove that $\lim_{r \to 1} \int_{-\pi}^{\pi} f(re^{i\theta}) d\theta = \int_{-\pi}^{\pi} f(e^{i\theta}) d\theta$

...if $f$ is continuous in an open set $\Omega$ containing the unit circle $T$. Is the proof something along the line of: $T$ is compact hence $\exists \epsilon > 0$ such that $D(z;\epsilon) ...
2
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1answer
46 views

Show limit exists of quotient of measures

This is a Theorem from Mattila's Book Geometry of sets and measures in Euclidean spaces: Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a separable metric space $X$. There ...
2
votes
0answers
37 views

Why is convergence of measures tested against functions?

This question is to help my intuition. Why do we test the convergence of measures against different classes of functions and not use definitions like: If $(B,\mathcal{B})$ is a measurable space then ...
2
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0answers
64 views

How to prove the sequence $f_n(x)=(\sin (nx))^n $ on the interval $[0,\pi]$ is not almost everywhere convergent?

How to prove the sequence $f_n(x)=(\sin (nx))^n$ on the interval $[0,\pi]$ is not almost everywhere convergent?
2
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0answers
67 views

Is f(A) Lebesgue measurable when A is lebesgue measurable and f is a function of the class C1? [duplicate]

Let A be a Lebesgue measurable set. Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be a function of the class $C^1$; Is this true that f(A) is lebesgue measurable? I know that this is true when f is ...
2
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0answers
34 views

Under what conditions on the experiment does bootstrapping work?

For a proof I would like to pretend that the uniform distribution on a finite set of samples from a 'source' eventually becomes the source's distribution a.s. when you keep adding samples. I am not ...
2
votes
1answer
70 views

Interchanging Inverse Laplace Transform

I have a function $f(|\boldsymbol{k}|,s,\theta)$ for which I am interested in its inverse Laplace transform. I am also interested in the function's mean value for constant $|\boldsymbol{k}|$, but ...
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0answers
26 views

$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n)$

I am trying to prove the equality $$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n),$$where $\mathcal B(\mathbb R^i)$ is the Borel $\sigma$-algebra on $\mathbb ...
2
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0answers
58 views

An exercise on conditional expectation and some related questions.

I tried to solve an exercise involving conditional expectations, and in doing so some question's popped up in my mind. First the exercise: $|Z| \le c \textrm{ P.-a.s.} \Rightarrow |E\{ Z | ...
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votes
0answers
67 views

Convergence of a subsequence of a subsequence of distribution functions

I'm trying to find a solution for the following problem: Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of signed (Baire)-measures (of bounded variation) on $[a,b]$ and let $F_{\mu_n}(t):=\mu_n([a,t))$ ...
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0answers
32 views

Product measure and splitting integral

Let $(X, A, \mu), (Y, B, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu)$, $g \in \mathcal L^1 (\nu)$. I want to show that $fg \in \mathcal L^1 (\mu \otimes \nu)$ $\int fg \ d(\mu \otimes ...
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0answers
36 views

Nonnegative function as limit of monotone increasing sequence (Measure Theory)

I am reading Bartle's "The Elements of Integration" and am at the part where he proves Lemma 2.11: If $f$ is a nonnegative function in $M(X,X)$, then there exists a sequence $(\phi_n )$ in $M(X,X)$ ...
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0answers
16 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
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0answers
51 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
2
votes
1answer
52 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
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0answers
39 views

Haar Measure on Locally Compact monoids

I have been reading on Haar measure and we know that every locally compact Hausdorff group admits a Haar measure, is the same true for semigroups with identity $e$(monoid)? If not, is there a class of ...
2
votes
1answer
82 views

Lebesgue measures defined on subspaces of $\Bbb R^n$

For any subspace $V$ of $\Bbb R^n$, we have a special measure $\lambda_V$ which can be described in various ways: Haar measure on $V$, or the measure induced by the metric $V$ inherits from $\Bbb ...
2
votes
2answers
51 views

Sequence of measurable functions $f_n=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$, uniform convergence

For each $n \in \mathbb N$, let $f_n:[0,\infty) \to \mathbb R: f_n(x)=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$. Show that there is no $E \subset [0,\infty)$ such that $|E|=0$ and $(f_n)_{n \geq 1}$ ...
2
votes
1answer
50 views

Show that $EX_1 1_A \geq 0$ given A is an event $\left(\sum\limits_1^n X_i>0 \right)$

Let A be the event $\left(\sum\limits_{i=1}^n X_i>0 \right)$ where $\{X_i\}_{i = 1, 2, ..., n}$ is an independent and identically distributed collection of random variables Show that $E[X_1 ...