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

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2
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1answer
39 views

Confusion about conditionals: $X\perp Y$ but $E\{Y|X\}\neq Y$? That's not right…

Protter Theorem 23.5. Let $Y$ be a positive or integrable r.v. on $(\Omega,\mathcal{F}, P)$. Let $\mathcal{G}$ be a sub $\sigma$-algebra. Then $E\{Y|\mathcal{G}\}=Y$ if and only if $Y$ is ...
2
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0answers
41 views

What is the link/ relationship and difference between probability measure, Bernoulli measure, Lebesgue measure, Borel measure and Hausdorff measure

I am having difficulty in understanding what is the difference between probability measure and Bernoulli measure. Is the latter used when the random variable has a Bernoulli distribution? What is its ...
2
votes
1answer
61 views

Continuity of the Fourier transform of a measure

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ then $$x \mapsto \hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle } \Bbb d \mu _{(y)}$$ is ...
2
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0answers
159 views

Constructing a Borel measurable function that assumes every real value in every sub-interval of (0,1)

So, this is a homework problem I have no idea how to approach. Previous year assignment sheets suggest a way which is as follows: Let $x ∈ (0, 1)$, and let $(x_i)_i≥1$ be its decimal expansion. (If ...
2
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0answers
34 views

Is it always true that $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$?

Suppose $\sum^{\infty}a_{i}1_{A_{i}}\geq \sum^{\infty}b_{i}1_{B_{i}}$, where $a_{i},b_{i}\geq 0$ and the sets possibly intersect i.e. $A_{i}\cap A_{j}\neq \varnothing $ and same with $B_{i}$. Is it ...
2
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0answers
30 views

If $(F_t)_t$ is a filtration, $T$ is a stopping time and $Y$ is $F_T$-measurable, then $1_{\left\{T=s\right\}}Y$ is $F_s$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal ...
2
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0answers
47 views

Distribution of density function $p(x)$ and $-\log p(x)$

Suppose $x$ is a random variable with a distribution that is absolutely continuous with respect to Lebesgue measure, such that it has a density $p(x)$. $p(x)$ is a function of a random variable, as is ...
2
votes
1answer
22 views

Criticism of proof attempt: Lebesgue convergence

Let $(f_j)_{j \in \mathbb{N} } \rightarrow f$ all be measurable, and positive functions ($f$ too). Assume $|f_n| \le f$ for all n. Show $\lim_{j \rightarrow \infty} \int f_j \ d \mu = \int f \ d \mu$. ...
2
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0answers
80 views

Rationale behind construction of measure theory from semirings

I am studying a book (Aliprantis & Burkinshaw, "Principles of Real Analysis") that, in order to introduce the concept of measure, starts from semiring. In particular the authors state that: ...
2
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0answers
35 views

Stein and Shakarchi, Exercise 2.25

Show that for each $\epsilon > 0$ the function $F(\xi) = \frac 1{(1+|\xi|^2)^\epsilon}$ is the Fourier transform of an $L^1$ function. [Hint: With $K_\delta(x)=e^{-\frac{\pi|x|^2}\delta} ...
2
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0answers
17 views

Characterization of Universal Measurablizers?

Ok, someone suggest a better title. Someone recently asked 1. Measurable sets : uncountable union, 2. null set is disjoint from translated rationals about how to show that $A+[-1,1]$ was Lebesgue ...
2
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0answers
27 views

How do the elements of the product-$\sigma$-algebra look like?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $\mathcal{E}$ denote he product-$\sigma$-algebra on $X$, where the set $\left\{0,1,2\right\}$ is equipped with the power set as $\sigma$-algebra. It ...
2
votes
1answer
87 views

Convergence of the characteristic function

Check that $$A_n \rightarrow A \iff 1_{A_n} \rightarrow 1_A$$ pointwise. Is my proof correct? Proof so far: $A_n \rightarrow A \iff \limsup_{n \rightarrow \infty} A_n=\liminf_{n \rightarrow ...
2
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0answers
67 views

Lebesgue decomposition theorem for measure on product space

Let's have two measurable spaces $(X,\Sigma_X),(Y,\Sigma_Y)$ and measure $\mu$ on $(X,\Sigma_X)$ and measure $\nu$ on product space $(X\times Y,\Sigma_{X\times Y})$. Does analogy of Lebesgue ...
2
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0answers
43 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
2
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0answers
146 views

Differentiation under the integral sign when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
2
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0answers
63 views

How $\sigma$-algebra determines random variable?

In my probability textbook there is a statement saying that Knowing the $\sigma$-algebra $\sigma(X)$ generated by a random variable $X$ is equivalent to knowing $X$ itself. We equate $\sigma(X)$ ...
2
votes
0answers
78 views

Closeness of measures on a cardinal

Given an uncountable $\kappa$ and a $\kappa$-complete nontrivial non-normal ultrafilter on $\kappa$, and some $g:\kappa\to\kappa$ with $<_{U}$-rank $\kappa$ (where $f_0<_Uf_1$ iff ...
2
votes
1answer
85 views

Repeated extension of Lebesgue measure

In Halmos' Measure Theory, section 16, exercise 2 deals with the extension of a $\sigma$-finite measure $\mu$ defined on a $\sigma$-ring $S$ to any set $M$ in the hereditary $\sigma$-ring induced by ...
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0answers
36 views

Example of a bounded simple process $A_t$ that changes value only once s.t. $\int_0^t A_s dB_s$ doesn't have normal distribution? [closed]

As the title of the question suggests, what is an example of a bounded simple process $A_t$ that changes value only once such that$$\int_0^t A_s\,dB_s$$does not have a normal distribution?
2
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0answers
71 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
57 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
votes
1answer
52 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
72 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
53 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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0answers
118 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
40 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 ...
2
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0answers
50 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
votes
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
95 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 ...
2
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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
38 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 ...
2
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0answers
50 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
99 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
votes
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
48 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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0answers
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
66 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
58 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
votes
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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0answers
42 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
143 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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0answers
67 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 ...
2
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0answers
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
votes
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 ...