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

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32 views

Proof monotone convergence theorem, why do they use this lim sup?

I have a question about the proof of the MCT. First they use a lemma, this is ok, but I'll show it for completeness: Now comes the proof. But I am wondering, why do they use a lim sup here?, why ...
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1answer
25 views

Product measurable functions

a) suppose $f,g:X\to[-\infty,\infty]$ are measurable. Prove the sets $$\{x:f(x)<g(x)\}, \ \{x:f(x)=g(x)\}$$ are measurables. I know that if $f,g$ measurables then ...
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1answer
55 views

Riesz representation theorem, uniqueness of the measure.

This question concerns theorem 2.14 (Riesz representation theorem) of Rudin's book, in particular, his claim that if $\mu_1,\mu_2$ are measures satisfying the hypothesis of the theorem then, ...
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2answers
53 views

Lebegue Dominated Convergence Theorem

Use the dominated convergence theorem to show that $$\lim_{n \to \infty} \int_{\mathbb{R}} (1+\frac{x^2}{n})^{-\frac{n+1}{2}} \mathrm dx = \int_{\mathbb{R}}e^{-\frac{x^2}{2}} \mathrm dx $$ You may ...
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1answer
68 views

Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Let $f(x)$ be a non-decreasing function on $[0, 1].$ You may assume that $f$ is differentiable almost everywhere. Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$. I am having a hard time with this ...
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0answers
44 views

Why are empty measurable spaces and empty topological spaces not desirable?

The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue ...
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2answers
70 views

More general definition of expected value

Let $X$ be a random variable with pdf $f$. I would like to know why: $$\operatorname{E} [X] = \int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)= \int_{-\infty}^\infty x f(x)\, ...
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1answer
42 views

Problem about $\sigma$-algebra

Space $\Omega$, $\mathcal C$ is a algebra,$\mathcal F=\sigma(\mathcal C)$ is a $\sigma$-algebra. define:$\mathcal F_\omega=\{B\in\mathcal F|\omega\in B\}$,$\mathcal C_\omega=\{B\in\mathcal ...
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0answers
14 views

Haar measure on locally compact semigroups

I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure? ...
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2answers
38 views

How do we prove the following set is measurable?

I was reading the proof of Egorov Theorem in the Real Analysis Book of Elias M Stein Suppose $\{f_k\}$ is a sequence of measurable functions defined on the measurable set $E$ with $m(E)< \infty$ ...
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0answers
31 views

Borel measure and positive linear forms

I'm just starting to learn about positive linear forms. If we call $C_{C}(X)$ the space of all continuous functions with compact support from domain $X$ and $\mathbb{C}$ (with $X$ a locally compact ...
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1answer
33 views

An inequality about signed measure.

Suppose $\mu$ is signed measure,then: $$|\mu(A)|\le\epsilon\Rightarrow|\mu|(A)\le2\epsilon$$ I tried to use the Jordan composition of $\mu$: $$\mu^+(C)=\mu(C\cap D),\mu^-(C)=-\mu(C\cap D^c)$$ so ...
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0answers
23 views

If $X$ is a LCHS and $f \in C_{C}(X)$ and $\mu$ is a Borel measure, then $f \in L^{1}(d\mu)$.

I want to prove the following statement: If $X$ is a locally compact Hausdorff topological space, and $f \in C_{C}(X)$ ($f$ is a continuous function with compact support), and if $\mu$ is a Borel ...
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2answers
49 views

A problem in Sigma algebra

I'm looking for ideas to solve the following problem: Let $(X,\mathbf{X})$ be a measurable space. If the $\sigma$-algebra $\mathbf{X}$ consists of a infinite number of subsets of $X$, then ...
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1answer
98 views

Equality about limsup.

Suppose $\sum_{n=1}^\infty \mathbb P(A_n)=\infty$,then: $$\limsup_{n\to\infty}\frac{(\sum_{k=1}^n \mathbb P(A_k))^2}{\sum_{i,k=1}^n\mathbb P(A_i\cap A_k)}=\limsup_{n\to\infty}\frac{\sum_{1\le ...
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1answer
66 views

How to understand the exchangeable $\sigma$-algebra?

Suppose there are $(\Omega,\mathcal F,\mathbb P)$ and r.v. $\xi_i$(i$\ge$1) $\xi_i:(\Omega,\mathcal F,\mathbb P)\to(\mathbb R,\mathcal B,\mu)$ $A\in$ the exchangeable $\sigma$-algebra $\mathcal E ...
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1answer
24 views

Adapted and backward adapted?

I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
2
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1answer
39 views

Example with almost every convergence where the dominated convergence theorem fails

So I ran into this exercise, and I want someone to check the accuracy of my answer, because I feel pretty sure that I make some mistakes which I can't see. Let $f_n(x) : \mathbb{R} \to \mathbb{R}, ...
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2answers
53 views

Show that inverse image of a Lebesgue measurable function is Lebesgue-measurable

I am struggling with this exercise. Can anyone please give me a hint? Suppose f is Lebesgue-Measurable. Show that $f^{-1}(B)$ is Lebesgue- measurable for any borel set B. I do know that both ...
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2answers
38 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
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2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
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1answer
10 views

Is a deterministic process adapted?

Let $B$ be a standard Brownian motion on a probability Space $(\Omega, \mathcal{F}, P)$ and let $\mathbb F:=(\mathcal{F}_t)_{t\in [0,T]}$ denote the natural filtration, i.e. $\mathcal{F}_t = ...
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1answer
52 views

A basic measure theory question

I have to prove that if $f$ is integrable on $\Bbb R^d$, real-valued, and $\int_E f(x)\, dx \geq 0$ for every measurable set $E$, then $f(x) \geq 0$ a.e. $x$. I don't understand where integrability ...
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1answer
24 views

Every analytic subset of $\Bbb{R}$ is the projection of a $G_\delta$ set $G \subset \Bbb{R} \times \Bbb{R}$

In the answer to this question (Projection of a set $G_\delta$, respectively in this post http://mathoverflow.net/questions/34142/projection-of-borel-set-from-r2-to-r1) it is claimed that every ...
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1answer
40 views

Do $\mathbb{R}^n$ and $\mathbb{C}^n$ valued ordinarily measureable functions form a Banach space under p-norm?

By measureable function I mean an "ordinarily" measureable function, that is measureable in a sense of this definition: a function between measurable spaces is said to be measurable if the preimage of ...
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1answer
14 views

Consistency of kernel density estimator with constant bandwidth

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = ...
3
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1answer
76 views

What does $\mathbb{P}(d\omega)=dw$ actually mean?

I am currently reading S. Shreve's book Stochastic Calculus II, and I have a question regarding Example 1.6.4 (p.35-36) which describes a change of measure, but I am puzzled by the notation. ...
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1answer
25 views

(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...
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1answer
39 views

Algebra generated by a collection of subset of a set

If I define algebra generated by a subset $S$ of power set of $X$ as intersection of all algebras containing $S$. Then is it true: every element in algebra generated by $S$ can be written as finite ...
2
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1answer
30 views

Caratheodory Measurability: Characterizations

Let $\mathcal{A}$ be an algebra over $X$ and $\mu:\mathcal{A}\to[0,\infty)$ a finite, positive and countably additive set function. Consider the induced outer measure: ...
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1answer
35 views

Lebesgue Outer Measure: Vitali Set

What is the Lebesgue outer measure of a Vitali set and its complement over $\Omega=[0,1]$? My first guess was zero and one but that was on my wrong idea that I can adjust the vitali set to lie within ...
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0answers
46 views

Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
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0answers
58 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
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2answers
46 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
2
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1answer
38 views

Conditional Expectation and Conditional Independence

Suppose we have 3 $\sigma$ algebras A, B and C such that A is independent of C. The random variable X is measurable with respect to $\sigma$(B,C), the $\sigma$ algebra generated by B and C. Is it true ...
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1answer
56 views

Question on verification of an example of regular conditional distribution

On page 197, Probability: Theory and Examples by Rick Durrett(See here), there's an example of regular conditional distribution Suppose $X$ and $Y$ have a joint density $f(x,y) > 0$. Let ...
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0answers
26 views

Radon-Nikodym: Complex Measures

Let $\Omega$ be a measureble space and $\mu$ a complex measure. (Note that this implies that the measure is finite.) Consider an absolutely continuous complex measure $\nu\ll\mu$. Then: $$\nu=\int ...
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0answers
37 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
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1answer
22 views

Problem EA 13.2 from David Williams' Probability with Martingales

I am stuck trying to solve this problem from Williams' Probability with Martingales: My attempt: $E(X_n) = E(e^{aS_n - bn})$ $= e^{-bn}E(e^{aS_n})$ (because $e^{-bn}$ is not random) $= ...
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1answer
19 views

Kernel density estimation in the limit of infinity many samples

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = ...
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1answer
59 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
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1answer
19 views

Is it sufficient to compare two measures on a generator?

Let $\mu_1$ and $\mu_2$ be two measures defined on a common sigma algebra $\Omega$, and let $\mathcal{G}$ be a non-trivial generator of $\Omega$. If $\mu_1(A) \leq \mu_2(A)$ for every element $A \in ...
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3answers
111 views

Volterra operator is completely continuous

Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by $$ Vf(t)=\int_0^t f d \mu $$ So, I want to show the following ...
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0answers
48 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
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2answers
41 views

Positive measure and $L^2$ space

I have a question about measure theory. Let $(E,\mathcal{B})$ be an arbitary measurable spase and let $m$ be a positive measure on $(E,\mathcal{B})$. Let $f \in L^{2}(E;m)$ (fix). $0 \leq\int ...
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1answer
89 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
2
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1answer
36 views

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are both complete measure spaces. Consider the following two measure spaces: $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ and $(X ...
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1answer
55 views

Caratheodory's Construction: Idea?

While reading Rudin's real and complex analysis I came across the following nice reasoning: Reasoning of Variation Measure Given a complex measure $\mu$ find its variation measure $|\mu|$ that is ...
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0answers
63 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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1answer
27 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...