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

learn more… | top users | synonyms (1)

0
votes
3answers
29 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
3
votes
1answer
16 views

Can this be proved using the MCT instead of the DCT?

I've seen various version of the DCT prove that if $f$ is a real valued, or extended real valued, or complex, integrable function, and if $\{E_n\}_n$ is a sequence of disjoint measurable subsets, ...
2
votes
1answer
23 views

Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following ...
1
vote
0answers
16 views

Explicit construction of Haar measure on a locally profinite group

Let $G$ be a locally profinite group. A Haar measure $\mu$ on $G$ is a measure defined on the $\sigma$-algebra $\mathcal B(G)$ of all Borel sets of $G$ with the following properties. 1) $\mu(K) ...
0
votes
1answer
26 views

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$.

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$. We know when $f$ and $g$ are both AC functions, the integration by parts is true. Is it ...
0
votes
0answers
18 views

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq ...
1
vote
1answer
13 views

Borel $\sigma$-algebra and natural number

Let $\Omega=\mathbb{R} $ and $\mathcal{S}=\{\{x\}:x \in \Omega \}$ a) $\mathbb{N} \in \sigma(\mathcal{S})$? b)Prove that $]0,1[ \not \in \sigma(\mathcal{S})$) ...
3
votes
2answers
44 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
0
votes
1answer
17 views

Borel isomorphism and approximation of Borel space valued function

In Kallenberg's Foundations of modern probability, he defines a Borel space $(S,\mathcal{S})$ as a measurable space which is Borel isomorphic to a Borel subset $B\in\mathcal{B}([0,1])$, ie., there ...
3
votes
0answers
44 views

Fubini theorem counter example

While reading Kallenberg's proof of Fubini theorem in Foundations of modern probability, I realized that he first proved Tonelli's theorem, then apply Tonelli to $f_+$ and $f_-$, the positive and ...
2
votes
1answer
44 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
33 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 ...
3
votes
2answers
46 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
21 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?
1
vote
1answer
49 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 ...
1
vote
0answers
22 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
43 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
51 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 ...
2
votes
0answers
32 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
36 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
57 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
20 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
93 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
36 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
14 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
35 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 ...
2
votes
1answer
39 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
75 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
21 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
34 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
43 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
62 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 ...