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

learn more… | top users | synonyms (1)

1
vote
1answer
31 views

On a step of a proof of the Borel-Cantelli lemma.

This is an excerpt taken form Probability with Martingales by Williams. The framework is probability theory. Why is the equation being discussed true if condition $\{ n \ge m \}$ is replaced by ...
4
votes
1answer
63 views

Subsets $B$ of bounded subinterval $I$ is lebesgue measurable iff $\lambda^*(I)=\lambda^*(B)+\lambda^*(I\cap B^c)$

Hi I was reading Cohn's book and I have problem with the following exercises (only the return of b is what I don't know), I'd appreciate any help and suggestion, if necessary, for a): a) Show ...
0
votes
0answers
19 views

Product measurable set induced by nonnegative function

Suppose $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f: X \rightarrow \mathbb{[0, \infty]}$ a nonnegative function. Let $G_f = \{(x,y) \in X \times [0, \infty]: y \leq f(x)\}$. ...
1
vote
1answer
42 views

Why do we need set $A$ to be countable in following proposition?

To define our terms: Let $\{ X_{\alpha} \}_{\alpha \in A } $ be any collection and $X = \prod_{\alpha \in A} X_{\alpha} $. Let $\pi_{\alpha}: X \to X_{\alpha} $ be coordinate maps and let ...
4
votes
1answer
71 views

Lebesgue measure of a subset of the unit circle

I'm having trouble getting started on this question: Let $S^1$ be the unit circle. Let $m = d \theta$ be the Lebesgue measure on $S^1$. Let $M \subset S^1$ be a measurable set such that $m(M) \geq 3 ...
2
votes
1answer
24 views

Using the dominated convergence theorem to bound the integral of a random variable

The following claim is used in the solution to problem 9.4 in Jacod and Protter's Probability Essentials: Claim: Let $X\in\mathcal{L}^{1}$ on $(\Omega,\mathcal{A},P)$ (where $\mathcal{A}$ is a ...
3
votes
1answer
41 views

The derivative of a measure

Let $\mu$, $\nu$ be two Radon Measure on $\mathbb{R}^n$. How can I prove that $D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))}$ is in $L^1_{loc}(\mathbb{R}^n,\mu)$?
2
votes
1answer
33 views

Showing a collection generates a sigma algebra

Let $X = X_1 \times X_2 $, let $\pi_i : X \to X_i $ ($i=1,2$) be coordinate maps. Let $\mathcal{M}_i $ be $\sigma-algebra $ on $X_i$. Let $\mathcal{F} = \{ \pi_i^{-1}(E_i) : E_i \in \mathcal{M}_i \} ...
1
vote
0answers
32 views

Definition of measurable functions whose values belong to a Banach space

Let $(X, \mathcal M, \mu)$ be a measure space and let $B$ be a Banach space over $\mathbb R$ or $\mathbb C$. It seems to me the most natural definition of a measurable function $f: X \rightarrow B$ is ...
0
votes
1answer
30 views

Existence of a Particular $L^1$ Function

I hope this question hasn't already been posted before in some other form (I couldn't find it, so if it has, please pardon me). I found this question on an old qualifier, but I am completely lost as ...
1
vote
2answers
68 views

porous sets: why measure zero?

We call a measurable set $E\subset\mathbb R^N$ porous if every ball $B_r(x)$ contains a smaller ball $B_{cr}(y)$ for some $c\in(0,1)$ such that $$ B_{cr}(y)\subset B_r(x)\setminus E. $$ So I've read ...
3
votes
3answers
71 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
1
vote
1answer
37 views

Is every continuous CDF the limiting distribution of some sequence of discrete CDFs?

Note: I know that for various measure-theoretic reasons (that I don't fully understand) this does NOT apply to the underlying probability density. I'll accept as answers either a proof, paper to a ...
1
vote
2answers
41 views

Do two almost surely equal random variables necessarily have the same probability?

Let $\Omega$ be a probability space with $\sigma$-algebra $\mathcal{A}$, and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $X:(\Omega,\mathcal{A}) \to (\mathbf{R},\mathcal{B})$ and ...
0
votes
0answers
27 views

Uniqueness of measure

I'm studying the Lebesgue-Besicovitch differentiation theorem (on the book "Sets of finite perimeter and geometrical variational problem- Maggi") but I'm not able to understand the following: Why the ...
1
vote
0answers
52 views

Example tripped Kolmogorov and Wiener

Assuming the hint is true, I attempt to prove the latter prop: Assume on the contrary that $\mathscr{L} = \mathscr{R}$. If $\sigma(Y_0) \subseteq \mathscr{L}$, then $\sigma(Y_0) \subseteq ...
2
votes
1answer
32 views

Convergence in measure theory context

I am studying for a test in measure theory. Please help with the following question: Let \begin{equation} f(x)=\begin{cases} \ln^2(x)& 0<x \\ 0,& x\leq 0\end{cases} \end{equation} For ...
-1
votes
2answers
35 views

Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$. [closed]

I am studying for a test in measure theory. Please help with the following question: Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$.
1
vote
1answer
32 views

Existence of minimal measurable majorant of an arbitrary function $f:X\rightarrow \bar{\mathbb{R}}$

Let $(X, \mathcal M, \mu)$ be a measure space. Let $f:X\rightarrow \bar{\mathbb{R}}$ be a function. A user wrote in his answer to this question(Lebesgue's monotone convergence theorem for upper ...
1
vote
0answers
36 views

Completion of a measure

Let $(X,\Sigma,\mu)$ be a measure space. Then, to complete the space, Rudin in his book Real and Complex Analysis considers $\Sigma ^\star$, the collection of all $E\subset X$ for which there exist ...
1
vote
1answer
83 views

Find Limit Using Lebesgue Dominated Convergence

I'm trying to find the following limits using Dominated Convergence Theorem, but can't seem to find a dominating function. Any guidance would be greatly appreciated! $\lim\limits ...
2
votes
0answers
125 views

Question on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
0
votes
2answers
20 views

Quesstion about outer measure definition

Reading "A First Look at Rigorous Probability Theory", and in the definition of outer measure of a set A, we take the infimum over the measure of covering sets for A from the semi-algebra (e.g., ...
1
vote
0answers
31 views

Vector valued measures

I can't understand what a vector valued measure is. In particular what does it mean to write $\nu (A)$ with A Borel measurable set?
3
votes
1answer
44 views

Locally constant property

Suppose f is positive and Schwartz function. Fix $N>0$ and $A>0$. Suppose that for any $x \in [-N,N]$, $$A \leq \int_{-N}^{N}f(x-z)dz$$ Then do the inequality $$A \leq C_{r} ...
0
votes
2answers
37 views

Prove that $ \frac {1}{f}$ is a function of bounded variation on $[a,b]$.

I am studying for a test in measure theory. Please help with the following question: Let $f:[a,b]\to R$ a continuous function of bounded variation, when $f(x)\ne 0$ for every $x \in [a,b]$. Prove ...
3
votes
2answers
88 views

Lebesgue's monotone convergence theorem for upper integrals

Let $(X, \mathcal A, \mu)$ be a measure space. Let $f:X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. It is sometimes useful to consider the so-called upper integral ...
1
vote
1answer
16 views

Sets formed from intersections of subsets of $\sigma$-algebras containing a fixed element

I'm working on the following problem. I know how to prove the desired result if you assume that $\mathcal{M}$ is countable (and thus finite. It is actually generally part of the proof that a countable ...
2
votes
0answers
50 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
7
votes
2answers
109 views

If $A_1 \subset A_2 \subset \mathbb R$ and $m^*(A_1) = m^*(A_2)$, will $m^*(A_1 \cap T) = m^*(A_2 \cap T), \forall T \subset \mathbb R$?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} l(I_n): E \subset ...
1
vote
1answer
31 views

Property of Radon measure on $\mathbb{R}^n$

Let $\mu$ a Radon measure on $\mathbb{R}^n$, $C$ a bounded and measurable set and $F_i$ a countable family of closed ball. I also have that $\frac{\mu(C)}{\eta(n)}\le \mu (C \cap \bigcup \{ \bar{B} ...
0
votes
1answer
50 views

On proving that a infinite intersection of truth sets is empty and on the usefulness of almost surely.

I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, ...
3
votes
0answers
73 views

Prove that $\lim\limits_{n \to \infty } \int_0^\infty (1 + x/n)^{-n}x^{-1/n}dx= 1$ using DCT

Prove that $\mathop {\lim }\limits_{n \to \infty } \int_0^\infty {\frac{{dx}}{{{{(1 + \frac{x}{n})}^n}{x^{\frac{1}{n}}}}}} = 1$ using dominated convergence theorem (DCT). By DCT we need to show ...
-4
votes
2answers
56 views

Prove that if $f^2$ is integrable in $X$, then $f$ is integrable in $X$.

I am studying for a test in measure theory. Please help with the following question: $(X,A,\mu)$ a finite measure space, and $f$ is a measurable function in $X$. Prove that if $f^2$ is integrable in ...
5
votes
2answers
76 views

A question about measure set

Suppose that a sequence of sets $\{A_n:n\in \Bbb N\}$ is increasing, and $A=\bigcup_{n=1}^\infty A_n$. If $A$ is measurable, $\mu(A)\gt 0$ and $\mu$ is an atomless measure, do there exist an $n\in ...
1
vote
1answer
30 views

Measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)h(t)$ given measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)$?

Suppose I know that, given $f(t), g(t) \in L^2(\Omega(t))$, $$t \mapsto \int_{\Omega(t)}f(t)g(t)$$ is measurable on $([0,T], Lebesgue) \to (\mathbb{R}, Borel)$. Suppose $h(t) \in L^\infty(\Omega(t))$ ...
5
votes
1answer
39 views

A sufficient condition for almost everywhere equality

Let $f,g:(0,\infty)\to \mathrm{R}$ be monotone decreasing functions. Show that if $m(\{x:f(x)>a\})=m(\{x;g(x)>a\}),\; \forall a\in \mathrm{R}$ where $m$ denotes Lebesgue measure, ...
2
votes
1answer
29 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
2
votes
2answers
110 views

Inclusions relating standard norms, in measury theory

I know that for finite measure space $(X, \mathcal A ,\mu )$ and $1\leq p< q<\infty $ , the inclusion $\mathcal L^q\subseteq \mathcal L^p\subseteq\mathcal L^1 $ holds true (applying Holder's ...
2
votes
1answer
57 views

On the equality of two sets (a doubt from Probability with Martingales).

Let $(S, \Sigma, \mu) $ be $([0,1], \mathcal{B}[0,1], Leb)$. Let $\epsilon(k)$ be a sequence of strictly positive numbers s.t. $\epsilon(k) \downarrow 0$. Let $V = Q \cap [0,1],$ the set of rationals ...
0
votes
2answers
48 views

lebesgue measure basic exercise

I have a basic question about a Lebesgue measure exercise that I am not sure how to solve. (I apologize if this is a simple question, I am new with this subject). Compute the Lebesgue measure of $X$ ...
5
votes
2answers
66 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
4
votes
5answers
74 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
4
votes
2answers
62 views

A problem on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
0
votes
1answer
35 views

Is the function $\ln (u(x))$ integrable when $u$ is bounded and positive?

Consider $\Omega$ an open bounded domain in $\mathbb{R}^n$ and $ u \in L^{\infty} (\Omega)$ a positive function. My question is : the well defined function $\ln (u(x))$ is integrable? Intuitively ...
8
votes
2answers
67 views

Lebesgue Measure of Image of Unit Square under Continuous Map

Problem. Let $h\in C(\mathbb{R})$ be a continuous function, and let $\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined by \begin{align*} ...
0
votes
0answers
39 views

finite signed measure on [0,1]

I'm studying for a qualifying exam and I'm stuck with the following question from an old exam; any help would be greatly appreciated: is there a finite signed measure $\mu$ on $[0,1]$ such that $ \int ...
3
votes
0answers
55 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...
7
votes
2answers
91 views

When is the union of $\sigma$-algebras atomless?

Suppose that we are given a probability space $(\Omega, \mathcal{F}, \mathsf P)$ and an increasing sequence of $$\mathcal{F}_1\subset \ldots \subset\mathcal{F}_n\subset \mathcal{F}_{n+1} \subset ...
3
votes
2answers
39 views

A weaker form of Lebesgue's differentiation theorem in $\Bbb R ^n$

If $f : \Bbb R ^n \to \Bbb C$ is locally-integrable then Lebesgue's differentiation theorem says that $$\lim \limits _{r \to 0} \frac 1 {\lambda \big( B(x, r) \big)} \int \limits _{B(x, r)} f \Bbb d ...