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

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2
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2answers
22 views

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
0
votes
0answers
16 views

Application of Carathéodory outer measure theorem

We know Carathéodory outer measure theorem and its proof, but I want an application or example about this theorem (I mean give me an outer measure and show the class $Μ$) I don't want this outer ...
2
votes
2answers
38 views

What does the conditional expectation look like when the $\sigma$-algebra is infinite

Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional ...
0
votes
1answer
22 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
1
vote
0answers
27 views

Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$ where the $f_n$ are measurable functions? ...
8
votes
2answers
105 views

Show that a given set has full measure or measure 0

Let $E \subseteq \mathbb R$ be Lebesgue measurable. And $E + q = E$ for any rational number $q$. Show that either $E$ or its complement has measure $0$. I tried this problem for few hours but ...
1
vote
1answer
44 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 infinite $\sigma$-algebra also generated by "kind of" partition? Can anyone help provide a explanation or ...
4
votes
2answers
83 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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0answers
26 views
2
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1answer
79 views

Question about notion $d\mu = fdv$ in Real Analysis of Folland

I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and ...
0
votes
0answers
23 views

Differentiability of parameter-dependent integrals 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 ...
0
votes
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 ...
1
vote
1answer
19 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 ...
1
vote
1answer
27 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 ...
1
vote
0answers
63 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 ...
0
votes
0answers
14 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
votes
2answers
311 views

Must a Borel set B of nonzero measure contain an interval as a subset?

Assume we are working in the reals under the standard Lebesgue measure. Must a Borel set B of nonzero measure contain an interval as a subset? I conjecture yes. Is the following line of reasoning ...
2
votes
0answers
36 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 $\sigma(X)$ ...
0
votes
1answer
33 views

Baire $\sigma$-algebra

Let $(X,T)$ hausdorff topological space. On the set $X$ it defines the baire $\sigma$-algebra, is written $B_0(X)=\sigma\{f; f \in C(X,\mathbb{R})\}$. Consider now the $\sigma$-algebra ...
4
votes
1answer
116 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 ...
0
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0answers
33 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 ...
7
votes
1answer
1k views

Proving the measure of an increasing sequence of measurable sets is the limit of the measures

Show that if $A_1\subseteq A_2\subseteq A_3\cdots$ is an increasing sequence of measurable sets(so $A_j\subseteq A_{j+1}$ for every positive integer $j$),then we have ...
5
votes
1answer
75 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
2
votes
1answer
22 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 ...
6
votes
1answer
443 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
1answer
21 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 ...
3
votes
3answers
482 views

Direct construction of Lebesgue measure

I have seen two books for measure theory, viz, Rudin's, and Lieb and Loss, "Analysis". Both use some kind of Riesz representation theorem machinery to construct Lebesgue measure. Is there a more ...
1
vote
1answer
46 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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votes
0answers
28 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 ...
3
votes
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 ...
1
vote
1answer
65 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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0answers
33 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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vote
0answers
28 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 ...
0
votes
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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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
vote
1answer
25 views

Expectation with respect to empirical distribution

Let $(\Omega,\mathcal{A})$ be a measure space and $X$ a random variable with distribution $P$. The expectation of some measurable function $g$ with respect to $P$ is $$ \mathbb{E}_P[g(X)] = ...
1
vote
1answer
31 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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votes
2answers
55 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?
2
votes
1answer
40 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 ...
1
vote
1answer
23 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 $\mu_g$. ...
2
votes
1answer
33 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?
0
votes
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 ...
1
vote
1answer
183 views

If the weighted $L^p$ norm of a measurable function is finite, is the weighted $L^p$ norm of the antiderivative also finite?

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} \,dx < \infty. $$ Define $g \colon \mathbb{R} \rightarrow ...
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votes
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 ...
5
votes
1answer
179 views

Show that the union over a collection of compact cubes in $\mathbb{R}^n$ is Lebesgue measurable

Let $\mathcal{K}$ be a (not necessarily countable) collection of compact cubes in $\mathbb{R}^n$. Show that $\cup\{K:K\in \mathcal{K}\}$ is a Lebesgue set (Measurable with respect to the Lebesgue ...
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votes
1answer
56 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
3answers
66 views

Example of disjoint union of sets which does not have additive measure

I had a question about the additivity property of the outer measure. Can someone provide an example of a disjoint union of sets which doesn't have an outer measure equal to the sum of the outer ...
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votes
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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1answer
39 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$, E = \bigcup_1^n{E_j}\}$ ...
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1answer
20 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 ...