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

learn more… | top users | synonyms (1)

1
vote
1answer
49 views

Measure of the reciprocal of a Cantor set

I have recently started studying measure theory and as is usual we started out by calculating the measure of the Cantor set. Now I had this question in my mind as to whether the set generated by ...
1
vote
1answer
25 views

Question about Haar Measure from Halmos

Halmos (Measure Theory, 1950, p. 256) poses the question: Given a Locally Compact Group $G$, compact subsets of measure zero, $C$ and $D$, is the group product, $P=CD$, which is also compact, also of ...
1
vote
1answer
47 views

Study the convergence of the sequence of functions $f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$ (convergence in measure, pointwise and in $ L^2(R ^d)$

Study the convergence of the sequence of functions $$f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$$ (convergence in measure, pointwise and in $ L^2(\mathbb{R} ^d)$). Let f be a measurable function such ...
1
vote
1answer
43 views

Differentiation theorem for Radon measures

I have trouble to understand a detail in the proof of the following Theorem: Theorem: Let $\nu, \mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be outer Radon measures, such that $\nu \ll \mu$. Then ...
1
vote
1answer
23 views

Need help with Lebesgue measures and continuity

Let $E\subset\Bbb{R}$ be a Lebesgue measurable set such that $\lambda(E)<\infty$ and $a,b\in\Bbb{R}$ such that $a<b$. (a) Prove that $f(x):=\lambda([a,x]\cap E)$ is continuous in $[a,b]$. (b) ...
1
vote
1answer
61 views

Help explain this “atom” and what this $G$ is.

Following are some slides from a lecture that you can watch here (starting at 20:40). It is explaining the Hierarchical Dirichlet Process. https://www.youtube.com/watch?v=PxgW3lOrj60 In the first ...
1
vote
1answer
71 views

Conditional Expectation of X

How do I calculate the conditional expectation of $E(X \mid X)$ where $E \vert X \vert<\infty$?
1
vote
1answer
62 views

Show that the distance $D_c$ between densities is symmetric when the densities are related by a linear transformation

The distance between two density functions $p_0$ and $p_1$ is given by $$D_c(p_0,p_1)=\int_{p_0/p_1>c} (p_0-c p_1)\mathrm{d}\mu$$ where $c>1$ is a real number Question: Show that if ...
1
vote
1answer
16 views

$\int\left|e^{i\langle s-t,x\rangle}-1\right|^2\mu(dx)=2(1-\Re(\varphi_\mu(s-t)))$ for all finite measures $\mu$ with $\mu(\mathbb{R}^n)=1$

Let $\mathcal{B}(E)$ denote the Borel $\sigma$-algebra on $\mathbb{R}^n$ $\mu :\mathcal{B}(\mathbb{R}^n)\to [0,\infty)$ be a measure with $\mu(\mathbb{R}^n)=1$ $\varphi$ be the characteristic ...
1
vote
1answer
179 views

Measure on the set of rationals

Consider the rational intervals defined as $$[a,b)_Q= \{r : a \leq r < b; r \in \mathbb{Q} \},a,b\in \mathbb{Q}, a<b. $$ Let $A$ be the class of all sets of rationals that can be produced as ...
1
vote
2answers
78 views

Definition of $\pi$ and d -systems.

Definition: Let $\Omega$ be a sample space. a) A d-system is a family of subsets containing $\Omega$ and closed under proper difference (if A,B $\in\mathcal D$ and A $\subseteq$ B, then B \ A ...
1
vote
1answer
45 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
1
vote
3answers
103 views

Difference of elements from measurable set contains open interval

Let $A\subset\mathbb{R}$ be a measurable set s.t $,m(A)>0$. Prove that the set $$B=\{x-y\mid x,y\in A\}$$contains nonempty open interval around 0. I thought to take an interval in $A$, ...
1
vote
1answer
27 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
1
vote
1answer
49 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
1
vote
2answers
48 views

Using the MCT to evaluate the integral of a series

I'm studying for my Measure Theory final and I've come across a question that I can't seem to find an answer for. For each $n \in \mathbb{N}$ set $E_n:=[n,2n]$ and let $f:\mathbb{R} \to \mathbb{R}$ ...
1
vote
1answer
113 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...
1
vote
1answer
25 views

intergral of lim inf and lim sup

Let $(f_n)$ be a sequence of numerical measurable functions on some measurable space and assume that there exists some $\mu$-intergrable $g$ such that for all $n\in\mathbb N$ the inequality $|f_n|\leq ...
1
vote
1answer
64 views

Absolutely continuous but not monotone

I don't want to comment on an old question, so I'm asking a new one. The question I'm referring to is Absolutely Continuous and Strictly Increasing on a Subinterval. Specifically, I'm concerned about ...
1
vote
1answer
39 views

How to calculate the closeness of a set of numbers?

Given a set of numbers, I would like to have a measure of how close they are to each other. I would like the calculated measure produces a single value. How could I achieve this?
1
vote
1answer
37 views

Measurable set contains a sequence

I found this question and didn't manage to extrapolate from the hint, could anyone help? Here's the question for the sake of completeness: Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: ...
1
vote
1answer
35 views

Equivalency in the elementary measure theory

Show that: $f\geqslant0$ and $\int f =0 $ $\Leftrightarrow$ $\mu$({$x$$\in$$X:$ $f($x$)>0$})=$0$ My idea: Let {$x$$\in$$X:$ $f($x$)>n$}=$E_{n}$ $\mu$({$x$$\in$$X:$ ...
1
vote
2answers
20 views

Show that for any $\mu$-measurable set $E$ we have $\mu(A\cap E) = \mu(B\cap E)$

Let $\mu$ be an outer measure on $X$. Let $A\subset X$ and assume there is a $\mu$-measurable set $B\supset A$ with $\mu(B)=\mu(A) <\infty$. Show that for any $\mu$-measurable set $E$ we have ...
1
vote
2answers
33 views

Sequence of Measurable Sets

Suppose that $ E $ is a measurable subset of $ R $. Suppose $ \left \{ E_{i} \right \} $ is sequence of measurable subsets of $ E $. For any $ x \in E $, there exist an $ N_{x} $ ...
1
vote
1answer
123 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
1
vote
1answer
33 views

Problem with topological proof about borel measures

It is given a finite Borel measure $\mu$ on a polish space $E$. The claim is that $\mu$ is then a regular measure. In the proof, it is shown that for any closed set $A$ it holds$$(1) \quad \mu(A) = ...
1
vote
1answer
75 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
1
vote
1answer
52 views

Proving a function is Lebesgue integrable

I need to prove that $$\frac{|x|^\alpha}{1+x^2}$$ is Lebesgue integrable for $\alpha \in [0,1)$ but I'm not sure how to do this. I first tried expanding this using the Taylor expansion to show it is ...
1
vote
1answer
77 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
1
vote
1answer
56 views

I want to show that $|f(x)|\le(Mf)(x)|$ at every Lebesgue point of $f$ if $f\in L^1(R^k)$

I want to show this, where $Mf$ is a maximal function, and I have attain $$Mf(x)-|f(x)|=\sup_{0\le r\le\infty}\frac{1}{B(x,r)}\int_{B(x,r)}(|f(y)|-|f(x)|)dm(y)$$ and I have no idea how to show that ...
1
vote
2answers
30 views

Convergence in a Measurable set

Let $E$ the set of all $x\in[0,2\pi]$ at which $\{\sin (nx)\}$ converges. This implies that $E$ is measurable? Thanks you all.
1
vote
2answers
32 views

given $E_1, E_2, E_3, …$ prove that the measure of {$x \in X :$ x belongs to infinite number of sets $E_k$} is $0$

Say I have a $\sigma$-algebra $\mathcal{A}$ over a set $X$ and a measure $\mu$. Let $E_1, E_2, E_3, .... \in \mathcal{A}$ such that $\sum_{k=1}^\infty \mu(E_k)$ < $\infty$. let B = {$x \in X ...
1
vote
1answer
34 views

Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

! (http://i.imgur.com/Zwt1m1n.png) I need to do the question at the top of this image. I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim ...
1
vote
1answer
32 views

Where does the following intuition about $G_\delta$ sets fail?

Where does the following reasoning that $\mathbb{Q}$ is supposedly a $G_\delta$ set fail? "Proof": $\mathbb Q$ may be covered by selecting open sets $O_n$ such that $m(O_n)<\frac{1}{n}$ for ...
1
vote
2answers
50 views

Integral Measures: Variation

Given a measure $\lambda\geq0$. Regard a real function $h:\Omega\to\mathbb{R}$ with $h\in\mathcal{L}$. Consider the real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation ...
1
vote
2answers
34 views

Showing Convergence in $L^p$ norms

Let $X$ be a finite measure space and $1\le p<\infty$ and $\{f_n\}$ be a sequence in $L^p(X)$ such that coverge to $f$ in $L^p(X)$ . If there exists constant $K$ such that for every $n\in ...
1
vote
2answers
94 views

Borel-Cantelli (proof and application)

Hi I was reading the second volume of the Tao's Analysis book and in one exercise he's asking for a proof of Borel-Cantelli If we have a sequence $s_n\in \Omega$ of measurable sets s.t. ...
1
vote
1answer
21 views

(Hints please) Prove that $\psi(E)=(\mu\times\lambda)(E)$ for every $E\in S\times T$.

Given that $(X,S,\mu)$ and $(Y,T,\lambda)$ are $\sigma$-finite measure spaces with the measure $\psi$ defined on $S\times T$ such that $\psi(A\times B)=\mu(A)\lambda(B)$ whenever $A\in S$ and $B\in ...
1
vote
2answers
41 views

Show that random walk is a random variable

I am working on this question. Suppose $\{X_n, n \ge 1\}$ are random variable on the probability space $(\Omega, \mathcal{B},P)$ and define the induced random walk by \begin{align*} S_0=0, \, ...
1
vote
1answer
35 views

$\mu$ is signed measure, and $f$ is integrable w.r.t. the total variation |$\mu$|. Show that $f$ is integrable w.r.t. $\mu^+$, $\mu^-$

Consider $\mu$ is signed measure, and $f$ is a real value integrable function w.r.t. the total variation |$\mu$|. Show that $f$ is integrable w.r.t. $\mu^+$, $\mu^-$ and $\int f d\mu= \int f d\mu^+ ...
1
vote
1answer
60 views

For a measure zero set $A$, the union $A\cup B$ has zero measure if and only if $B$ does

Definition: A set $A$ has measure $0$ iff $\forall \epsilon > 0, \exists$ system of intervals $(I_\tau): A \subseteq \cup_\tau (I_\tau), 0 \leq \sum_\tau (\operatorname{length}(I_\tau)) < ...
1
vote
1answer
166 views

Completion of Lebesgue Measures

Hello Mathematics Community. I was hoping someone could assist me in solving the following problem from Terrence Tao's Introduction to Measure Theory book. I am using the free online version of the ...
1
vote
1answer
60 views

A question on Hausdorff measure

This question is part of a homework assignment. Considering the hausdorff measure $\mathcal{H}_{2}$ on $\mathbb{R}^{3}$, I need to compute the measure of the unit cube: $A = \{(x,y,z) \in ...
1
vote
1answer
71 views

Equal in distribution but unequal almost everywhere?

If this question has been asked, I apologize but I could not find it. I was wondering if it was possible construct $X$, $Y$ two iid rv's such that they equal in distribution, i.e. $P_X(B) = ...
1
vote
1answer
57 views

About generated $\sigma$-algebras (proof verification).

It would be really helpful if anyone would browse through this and tell me if my solution is ok. Here is the question: Let $C \subset 2^X$ be a collection of subsets. Show that for every $K \in ...
1
vote
1answer
17 views

Distribution of random variable $Y$ passed throught distributin function of $X$

If \begin{align*} F(x)=P[X \le x] \end{align*} is continuous in $x$, show that $Y=F(X)$ is measurable and that $Y$ has uniform distribution \begin{align*} P[Y \le y]=y, \, 0 \le y \le 1. ...
1
vote
2answers
65 views

A measurable piecewise function

I want to show that the following functions is measurable: $f:\mathbb{R}\rightarrow \mathbb{R}, f(x) = \begin{cases} \frac{1}{\sqrt {(1-x^2)}} & ,\text{if } x \in [-1,1] \\ 0 & ...
1
vote
1answer
165 views

Outer measure is not finitely additive

I know similar questions have been asked before, but I'm looking for clarification of a proof. In Royden's book on real analysis, he proves that every set of positive measure contains a non-measurable ...
1
vote
2answers
43 views

if $f \ge 0$ and $\int fd\mu<\infty$, then for any $a>0$ the set $\{f\ge a\}$ has finite $\mu$-measure

Let $(X,\Sigma , \mu)$ be a measure space. Show that if $f \ge 0$ and $\int fd\mu<\infty$, then for any $a>0$ the set $E_a:=\{f\ge a\}$ has finite $\mu$-measure. My attempt: We know that ...
1
vote
1answer
34 views

How to prove that the sequence of random variables converges to a random variable?

If $Z_1,Z_2,\cdots,Z_n$ are random variables such that $Z(\omega)=\lim_{n \to \infty}Z_n(\omega)$ exists $\forall \omega \in \Omega$, then $Z$ is also a random variable. I was reading a book on ...