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

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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
123 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
63 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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2answers
147 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
64 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
84 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
26 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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230 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
117 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
199 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
40 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 $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
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1answer
98 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. Let $f_n(x) : \mathbb{R} \to \mathbb{R}, f_n(x)= n\mathcal{X}_{[0,\frac{2}{n}]} \forall$ n $\in \mathbb{N} $. ...
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2answers
237 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
53 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
68 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
244 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
62 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
133 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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59 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
67 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) = ...
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1answer
193 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
35 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
53 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 ...
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1answer
134 views

Caratheodory: Measurability

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
183 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
70 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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96 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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3answers
158 views

Bochner Integral: Integrability

Attention This question has been slightly modified!! Reference It is related to: Bochner Integral: Axioms Problem Given a measure space $\Omega$ and a Banach space $E$. Consider Bochner ...
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1answer
240 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
106 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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146 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
288 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
53 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
88 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
28 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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2answers
253 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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1answer
292 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
45 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
118 views

Does the symmetric decreasing rearrangement of a smooth function preserve smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
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1answer
188 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 ...
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1answer
69 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
123 views

Caratheodory: Motivation

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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161 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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81 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 $ ...
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109 views

Set of measure zero

Let $\mathcal{S}$ be the set $\mathcal{S} = \{(\mathbf{x}, \mathbf{y}) \in \mathbb{C}^{n} \times \mathbb{C}^{n} \mid \mathbf{x}^{H}\mathbf{y} = ||\mathbf{y}||^{2}_{2}\}$. Does $\mathcal{S}$ a set of ...
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22 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
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1answer
89 views

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem. I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$. Is ...
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35 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
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1answer
43 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
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0answers
117 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...