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

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22 views

Is every $\sigma$-algebra generated by a partition?

I know that every finite $\sigma$-algebra is generated by a finite partition, but is every countable $\sigma$-algebra also generated by a countable partition? How about uncountable $\sigma$-algebra? ...
1
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1answer
13 views

Additive set function on a semiring of sets

A semiring $\Pi$ on a set $X$ is a non-empty family of subsets of $X$ with the following properties. 1) $P \cap Q \in \Pi$ whenever $P\in \Pi$ and $Q\in\Pi$. 2) $P - Q$ is a finite disjoint ...
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0answers
10 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
4
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2answers
65 views

Why is $(\mathbb{R}, \mathcal{P}(\mathbb{R}))$ called a measurable space when actually is not?

I get confused when I put the following three notes together: Power set of any set is a $\sigma$-algebra. If $X$ is a set and $\Sigma$ is a $\sigma$-algebra over $X$, then the pair $(X, \Sigma)$ is ...
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1answer
21 views

question on measurability of a function

Let us define a function $f:[0,1]\to \ell_\infty[0,1]$ by $f(t)=\chi_{[0,t]}$. Here $\ell_\infty[0,1]$ stands for the space of all bounded functions from $[0,1]$ to $\mathbb{R}$, where $[0,1]$ is ...
2
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0answers
25 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 $\sigma(X)$ ...
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0answers
25 views

Exercise 3.32 from Real Analysis of Folland

Can someone give me some hint on how to solve this problem? Thanks a lot If $F_1, F_2, ..., F \in NBV$ and $F_j \rightarrow F$ pointwise, then $T_F \le \liminf T_{F_j}$ Here, NBV is the ...
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1answer
19 views

Does weak-$\ast$ convergence with an exponential rate imply convergence of measures of sets with the same rate?

Assume that $\mu_n \to \mu$ in the weak-$\ast$ topology with the following rate for any compactly supported continuous function $f$: $$|\mu_n(f) - \mu(f)| \leq C_f e^{-n}.$$ Can we replace $f$ with ...
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0answers
27 views

Understanding the set structure of probability theory [on hold]

Since events have their own probabilities and outcomes have their own probabilities. Why don't we just consider only one of events or outcomes directly? What's the motivation to have this set-point ...
1
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0answers
26 views

Borel $\sigma$-field and Equality

Let $\mathcal{B}$ be a Borel $\sigma$-field on $\mathbb{R}$ and let $\mathcal{C}$ be the collection of closed intervals on $\mathbb{R}$. Show that $\mathcal{B} = \sigma(\mathcal{C})$. If I'm going ...
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1answer
43 views

A question about 2.1 Proposition on Folland's Real Analysis

Definition of measurable space: If $X$ is a set and $\mathcal{M} \subset \mathcal{P}(X)$(Power set of $X$) is a $\sigma$-algebra, $(X, \mathcal{M})$ is called a measurable space and the ...
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0answers
12 views

How to deduce this fact from the existence of factorized regular conditional probabilities and disintegration of probability measures?

In the lecture we had a theorem about the disintegration of probability measures in the following formulation: Theorem: Given two standard Borel spaces $(S_i,\mathscr S_i),i=1,2$ let $(S,\mathscr ...
1
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1answer
30 views

Mean value formula integrals

Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Then I was wondering whether $$\frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) dS(x) \rightarrow_{r ...
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0answers
30 views

Equivalent Definition of Weak $L^{p}$ (Quasi-) Norm

For a sigma-finite measure space $(X,\Sigma,\mu)$, the weak $L^p$ (hereafter denoted $L^{p,\infty}$) is defined by $$\|f\|_{L^{p,\infty}}:=\sup_{t>0}t\mu(|f|>t)^{1/p}, \qquad (1\leq ...
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0answers
53 views

Explicit construction of Haar measure on a profinite group

Let $G$ be a profinite group. It is known that in $G$, every neighborhood of the identity element contains an open compact subgroup. I would like to explicitly construct the Haar measure on $G$. The ...
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0answers
20 views

What is the interpretation of $\nu(dy - x)$ where $\nu$ is a Lévy measure?

In a paper I am reading, it is seemingly suggested that, if $\nu(dx)$ is a Lévy measure, then the following holds for a function $f(x)$ which is smooth (and satsifies some integrability conditions): ...
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2answers
54 views

intuition of mass function of random variable [on hold]

When we are using $P\{X=x\}$ it seems like intuitively there is a function from $T$ (or measure from $\mathcal{B}(T)$) to $[0,1]$. What is the theoretical foundation behind this intuition?
3
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1answer
42 views

Understanding product $\sigma$-algebra

Let $\{X_\alpha\}_{\alpha \in A}$ be an indexed collection of nonempty sets, $X = \prod _{\alpha \in A}X_\alpha$, and $\pi _\alpha: X \rightarrow X_\alpha$ the coordinate maps. If $M_\alpha$ is a ...
2
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1answer
32 views

Class of subsets which is not a $\sigma$-ring

I can't find a non-empty class that is closed under countable intersections and symmetric differences, but it's not a $\sigma$-ring. Any ideas?
2
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1answer
38 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
0
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1answer
27 views

Is there a Markov-type inequality for the Median?

Markov's theorem states that $P(|X| \geq a) \leq \frac{E[|X|]}{a}$. Is there an similar type of inequality that involves the median (somehow I doub't it, but I make no claim to comprehensive knowledge ...
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0answers
22 views

Is there a difference between $\mu_1 \times \mu_2$ and $\mu_1 \otimes \mu_2$ in measure theory?

I sometime see the tensor product symbol used when referring to product meauress, but I've also seen the cartesian product symbol used too. Is there a difference. I have had a hard time finding an ...
1
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1answer
22 views

Volume density on a Riemannian manifold as a measure

I am having some trouble in seeing exactly how the Riemannian density form gives rise to a measure on $\text{Borel(M)}$. Let $(M,g)$ be a Riemannian manifold. We have the Riemannian density $\mu_g$. ...
1
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1answer
20 views

Sets cut into two halves of equal size by any straight line through a particular point

Is there an easy characterization of all sets $M \subseteq \mathbb{R}^2$ with the following property? A point $(x_M,y_M)$ (which may depend on $M$) exists such that each straight line through ...
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1answer
52 views

How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
4
votes
1answer
113 views

Is $\overline{D}_{\varepsilon}$ a connected Jordan region in $\mathbb{R}^{n}?$

Definition. Let $E$ be a nonempty subset of $\mathbb{R}^{n}$.The distance from a point $\mathbb{x}\in\mathbb{R}^{n}$ to set $E$ is defined by ...
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1answer
37 views

Question about formula for total variation of complex measure from Real Analysis of Folland

Let $\nu$ be a complex measure on $(X, \mathcal{M})$. If $E \in \mathcal{M}$, define: $\mu_1(E) = \sup\{\sum_1^n{|v(E_j)|}:n \in N, E_1, ..., E_n$ disjoint$, E = \bigcup_1^n{E_j}\}$ ...
1
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1answer
45 views

Property of a set of a positive Lebesgue measure

I am trying to see whether it is true that in any set of a positive Lebesgue measure in $R^2$ we can always find two points $(a_1,a_2)$ and $(b_1,b_2)$ such that the following hold: $a_1>b_1$ ...
2
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0answers
63 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 ...
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1answer
41 views

Jensen's inequality problem [on hold]

I want to know an example of a infinite measure space $(\Omega, \mathcal{F},\mu)$, real valued function $g$ and convex function $\phi$ defined on the real line s.t. $$\phi\left(\int g d\mu\right) > ...
1
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1answer
19 views

Predictability of $\int^t_0 f(X_s)\,\mathrm ds$ where $X$ is a Lévy process

Let $X_t$ be a Lévy process and $f(x)$ some smooth function. Under what conditions is $$ Y_t = \int^t_0 f(X_s)\,\mathrm ds$$ predictable? Not sure how to investigate this. It is clearly adapted, so ...
1
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1answer
37 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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2answers
42 views

$\sigma$-algebra of $\mathbb{R}$ generated by $\mathcal{P}(\mathbb{N})$

What is the $\sigma$-algebra of $\mathbb{R}$ generated by $\mathcal{P}(\mathbb{N})$? I thought it is $$\Sigma = \{\emptyset, \mathbb{N}, \mathcal{P}(\mathbb{N}), \mathbb{R}, \mathbb{R}-\mathbb{N}, ...
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0answers
37 views

Generating structure of Borel field

On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the author wrote: ...and there are Borel sets that cannot be arrived at from the intervals by any finite sequence of set-theoretic ...
2
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0answers
23 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?

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?
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1answer
29 views

convolution of probability measures

What do we mean by convolution of measures? With example What is the difference between convolution of measures and convolution of functions? What is probability measure? Give an example of ...
2
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1answer
34 views

Measurability of marginal distributions of a random measurable function

For a probability space $(\Omega, \mathcal F, \mathsf P)$, let $X \colon \Omega \times [0,1] \to \mathbf R \colon (\omega, t) \mapsto X(\omega,t)$ be a random Borel function (i.e. an $(\mathcal ...
1
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1answer
35 views

Properties of decreasing sequence of Lebesgue measurable sets.

I'm trying to prove a property of Lebesgue measure sets that says: If the $A_{k}$'s are measurable and $A_{1} \supset A_{2} \supset A_{3} \supset \ldots,$ and if $\lambda (A_{1}) < \infty, $ then ...
2
votes
1answer
24 views

Help with a sigma-algebra problem with random variables (show $\sigma(X_S)\subseteq \sigma(X_T)$ if $S\subseteq T$)

My problem is as follows: Let $X_S$ and $X_T$ be two stochastic processes where $S,T$ are index sets. Let $\sigma(X_S)$ and $\sigma(X_T)$ denote the sigma-algebra generated by $X_S$ and $X_T$. ...
7
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0answers
65 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
2
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0answers
40 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

Lebsegue measure of $\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \}$

Find the Lebsegue measure of the set $A= \left\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \right\}$. The answer given is $1 - \ln \sqrt{2}$. My thought: I only know that Lebsegue ...
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0answers
15 views

What is the difference between a “Borel probability measure” and a “singular Borel probability measure”? [closed]

What is the difference between a "Borel probability measure" and a "singular Borel probability measure"? When a probability meausure is said singular? Thanks in advance.
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0answers
22 views

Explicit constructions of Haar measures?

I know how to build the Haar measure somewhat explicitely on Lie groups (via differential forms) and profinite groups (by using the lemma that open subsets of a profinite group are unions of cosets of ...
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0answers
32 views

Basic Set-Theoretic Properties from Halmos

I've been backtracking lately to make sure that I have a solid set-theoretic background before taking measure theory this fall. Here's a few facts I've come across today, and my attempted proofs. Let ...
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0answers
31 views

Premeasure on $\mathcal{A}$ and $\mu^{*}$ proof

This proposition comes from Real Analysis by Folland: Some background information: (1.10) Let $\epsilon\subset P(X)$ and $p:\epsilon\rightarrow [0,\infty]$ be such that $\emptyset\in\epsilon$, ...
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0answers
32 views

Countable $\sigma$-algebra [duplicate]

Let $\Sigma$ be a countable $\sigma$-algebra. Show that there is a sequence $A_1,A_2,...$ of disjoint elements of $\Sigma$ such that every $B$ in $\Sigma$ is a countable union of elements in ...
2
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1answer
29 views

Constructing dependent sequences of random variables

It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. ...
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1answer
51 views

Easy proof for existence of Lebesgue-premeasure

In the lecture on measure theory I attended last semester, we had a sort of complicated technical proof for the existence of the Lebesgue-premeasure. However, I can't see why this easier argument does ...
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0answers
46 views

$f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [duplicate]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...