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

learn more… | top users | synonyms (1)

1
vote
1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
0
votes
2answers
41 views

$L^{\infty} (X, \mu)$ is not separable? [closed]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
2
votes
1answer
19 views

Does simply-connected imply measureable?

The famous examples of non-connected sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space is ...
0
votes
1answer
31 views

open set $O$ such that $\partial(\overline{O})$ has positive measure

Find an open set $O$ such that $\partial(\overline{O})$ has positive measure. The hint is to consider a Cantor set, with positive measure. But that does not work, because all the Cantors are closed ...
1
vote
1answer
19 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
0
votes
0answers
27 views

Borel image of the projection [closed]

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps Borel sets to Borel sets?
5
votes
1answer
132 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
votes
1answer
36 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
0
votes
2answers
26 views

A proposition about positive random variables and expected values

I have problems to give a proof for the following proposition: Consider a random variable $X$ with values in $[0,+\infty]$. If $P(X=+\infty)>0$, then $E(X)=+\infty$ (notation: $E(X)=\int X ...
4
votes
2answers
221 views

Is this set measurable?

Let $E$ be a subset of $\mathbb{R}$. Assume that $\forall x\in E, x$ is a limit point of $E\setminus\{x\}$. Then, is $E$ Lebesgue-measurable? For example, any perfect subset, open subset or ...
1
vote
1answer
54 views

Confusion with real numbers and random variables; Integration and Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let $X_n$ be iid RVs with the same continuous dist function. Let $E_1 = \Omega$ and for $n \geq 2, E_n = (X_n > X_m ...
2
votes
1answer
63 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
0
votes
1answer
25 views

Probability of highest common factor in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let s > 1 and let $\zeta(s) = \sum_{n=1}^{\infty} {n^{-s}}$. Let X and Y be independent $\mathbb{N}$-valued random variables ...
6
votes
1answer
60 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
1
vote
0answers
43 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
1
vote
0answers
19 views

Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
1
vote
0answers
24 views

Completely stumped on exercise cooncerning the characterisation of Jordan measure - would anyone be willing to give a hint?

In Terry Tao's notes on measure theory he has the following exercise, I have no idea how to deal with the last statement, I would really appreciate it if someone could give a hint for the final case. ...
1
vote
1answer
21 views

Conditions for “forward” measure-preservation

A transformation $T$ being $\mu$-invariant is by definition a transformation satisfying $$\mu(T^{-1} E) = \mu(E)$$ for all measurable sets $E$. I was wondering what are sufficient conditions for being ...
1
vote
0answers
31 views

Independence Exercise in Rosenthal

In Rosenthal's, "A First Look At Rigorous Probability Theory", $\exists$ this exercise: Exercise 3.6.19. Let $A_1,\ A_2,\ldots$ be independent events. Let $Y$ be a  random variable which is ...
5
votes
4answers
101 views

Convergence in measure implies pointwise convergence?

In showing that we can replace pointwise convergence with convergence in measure in the Lebesgue Dominated Convergence Theorem, I made the following claim: 1.) $f_n\to f$ in measure ...
0
votes
1answer
37 views

Proof that $X$ countable, $\mathcal M$ algebra on $X$ implies $\mathcal M$ a $\sigma$-algebra

In my measure theory class, I believe the professor made the claim that if $X$ was a countable or finite set and $\mathcal M$ was an algebra on $X$, then $ \mathcal M$ was a $\sigma$-algebra. I am ...
2
votes
1answer
30 views

Independence of Events in Rosenthal

$\exists$ this exercise in Rosenthal's A First Look at Rigorous Probability Theory: For letters d and e, how do you show that the ff events are independent? My attempt: It suffices to show that ...
1
vote
1answer
25 views

Question on a variation of Borel Cantelli Lemma

In this question, what is the purpose of the summation? If the limit of the sequence is zero, the corresponding series is convergent. Does the desired conclusion not then follow from BC1?
1
vote
1answer
32 views

Continuity of measure and integration

Suppose that f is a measurable function $(\Omega, \mathfrak{F}, \mu)$ such that $\int_{A}f \, d\mu \geq 0 \forall A \in \mathfrak{F}$. Prove that $f \geq 0 \ \mu$-almost surely. Hint: Let $A_n = ...
0
votes
1answer
54 views

Using Borel-Cantelli Lemma

Let $X_1, X_2,\ldots$ be iid Geometric(p) where $p \in (0,1)$. Thus if $q=1-p$, then $P(X_n > k) = q^k$ for $k\geq 0$. Prove that for any fixed $\epsilon \in (0,1)$, CORRECTION: k is supposed to ...
-1
votes
0answers
35 views

Possion integral for measure dominated by the maximal function of the measure [closed]

Recently, I was thinking a problem about Possion integral for measure dominated by the maximal function of the measure, that is to say, let $\mu$ be a regular Borel measure in $\mathbb{R}$, define its ...
1
vote
2answers
58 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
2
votes
1answer
38 views

Do the subsets of $\mathbb N$ that have asymptotic density form an algebra?

Consider $ \Omega=\mathbb{N}.$ Is said that a $E\subset\mathbb{N}$ has a density limit if the following limit exists: $$\rho(E)=\displaystyle \lim_{n\rightarrow \infty} \dfrac{\#\{k\in E: k\leq ...
3
votes
1answer
56 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
2
votes
1answer
27 views

“Structure of a measure space is the coarsest among all substantial structures on a set…”

In the book Lectures and Exercises on Functional Analysis by Helemskii I have stumbled upon the following note: The Rohlin theorem and similar results (see e.g., [19],[20]) show that the structure ...
1
vote
0answers
22 views

Absolute continuity for non-measures?

Let $B$ be the collection of Borel subsets of $R^2$. A measure on $B$ is said to be absolutely continuous with respect to area if any subset with area 0 has measure 0. Is there a natural ...
0
votes
1answer
15 views

Absolute continuity of two-dimensional measures

Absolute continuity has two different meanings: one for functions and one for measures. The Wikipedia page explains the relation between the two notions in the following way: A finite measure μ ...
0
votes
1answer
22 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
1
vote
0answers
122 views

Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1} = \{x \in \mathbb{R}^d:|x|=1\}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every rotation $r$ of $\mathbb{R}^d$ and ...
2
votes
1answer
38 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
1
vote
4answers
139 views

Definition of a measurable function?

Are these two definitions of a real-valued, measurable function equivalent? ($(X, \Sigma, \mu)$ is the measure space.) Definition 1: $f: X \to \mathbb{R}$ is said to be measurable if for all ...
2
votes
0answers
36 views

Characterisation of absolutely continuous measure on the real line

Let $\lambda, \nu$ be two Radon measures on $\Bbb R$ such that $\lambda(\Bbb R)< \infty$. Show that the following are equivalent: $\lambda \ll \nu$; $\forall \epsilon>0$ there exists ...
4
votes
1answer
69 views

Characterization of measurability by closed sets.

If $E \subseteq \Bbb R$ is measurable, then for all $\epsilon > 0$, exists $F \subseteq \Bbb R$ closed such that $F \subseteq E$ and ${\frak m}^*(E \setminus F) < \epsilon$. I have already ...
3
votes
1answer
54 views

Symmetric Borel sets in the plane

How will I show that the sigma algebra consisting of all Borel sets in the plane, which are symmetric about the line $y=x$, is generated by sets of the form $(a,b) \times (a,b)$? I could show upto the ...
4
votes
1answer
42 views

How to prove the following defined collection is a sigma algebra?

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
2
votes
1answer
45 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
0
votes
1answer
23 views

Probability densities and Absolute continuity

I've not deep knowledge in measure theory/real analysis but just few concepts given me during this second year probability course. I'm trying by myself to understand more, but I don't want to dive in ...
4
votes
2answers
142 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
0
votes
0answers
12 views

Folland's proof of the Hahn Decomposition. Minor error?

Theorem 3.3 of Folland's Real Analysis (ed 2) is the Hahn decomposition theorem. In the proof he assumes that the signed measure $\nu$ he is considering does not take the value $-\infty$. Then he ...
2
votes
1answer
22 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
2
votes
2answers
44 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
0
votes
1answer
45 views

sigmal algebra and measure [closed]

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
0
votes
0answers
23 views

Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
0
votes
1answer
18 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
4
votes
1answer
75 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...