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

learn more… | top users | synonyms (1)

2
votes
2answers
41 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
0
votes
0answers
25 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
2
votes
2answers
35 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
0
votes
0answers
20 views

Does a proper compact subset of a compact subset of R has strictly smaller measure?

Let K be a compact subset of R and H a proper compact subset of K. Does H has a strictly smaller Lebesgue measure than K?
0
votes
1answer
26 views

Use dominated convergence theorem to show:

$\lim_{n \rightarrow \infty} \int_0^{\infty}f_n(x)dx = \int_0^{\infty} \frac{x}{e^x-1}dx$, where $f_n(x):=\frac{n}{e^x-1}\sin\frac{x}{n}$ Hi I'm working on some practice questions and having a bit of ...
0
votes
0answers
18 views

Proving that a measure is continuous from below

Let $(X, \mathcal{M}, \mu)$ be a measure space and $\{E_j\}_{j=1}^\infty \subset \mathcal{M}$ such that $E_1 \subset E_2 \dots $ I want to prove that $\mu(\cup _1^\infty E_j) = \lim_{j \to ...
1
vote
2answers
39 views

Definition of $f \vee g$ and $f \wedge g$

In Olav Kallenberg's Foundations of Modern Probability he uses the notation $f \vee g$ and $f \wedge g$ where $f, g$ are two functions from a set $\Omega$ to $\mathbb{R}$. What does this notation ...
4
votes
2answers
50 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 $\mathcal{B}$ ?
2
votes
2answers
48 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
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 ...
1
vote
0answers
29 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? ...
-1
votes
0answers
27 views

Is continuous random variable always an “onto function” [on hold]

Is continuous random variable always an "onto function"? If yes, why?
0
votes
1answer
23 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?
0
votes
0answers
27 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 ...
2
votes
2answers
53 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 ...
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 ...
0
votes
0answers
15 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
90 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 ...
1
vote
1answer
28 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
votes
0answers
39 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
0answers
35 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 ...
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 ...
-1
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 ...
1
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 ...
1
vote
1answer
47 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 ...
0
votes
0answers
13 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
34 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 ...
1
vote
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 ...
1
vote
0answers
65 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
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): ...
-1
votes
2answers
56 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
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 ...
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?
2
votes
1answer
42 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
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 ...
0
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 ...
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
23 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 ...
-1
votes
1answer
59 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
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 ...
1
vote
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$, E = \bigcup_1^n{E_j}\}$ ...
1
vote
1answer
57 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
votes
0answers
71 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 ...
0
votes
1answer
42 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
vote
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 ...
1
vote
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 ...
1
vote
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}, ...
1
vote
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
votes
0answers
24 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? [on hold]

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?
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 ...