Tagged Questions

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

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3
votes
1answer
44 views

On clarifying the relationship between distribution functions in measure theory and probability theory

I recently found myself confusing concepts from measure theory and probability theory, so I'd like to get an idea for what I'm misunderstanding. This definition is what started it all: A sequence ...
0
votes
2answers
41 views

Find a subset of the real numbers

I have to find an open and dense subset of the real numbers with arbitrarily small measure. Since the set of the rational numbers is dense, could we use a subset of the rationals?? How could I find ...
0
votes
1answer
16 views

Borel algebra on the postive real line

I´m considering the borel sigma algebra on the positive real line, $ \mathcal{B} (\mathbb{R}_+ ) $ and I would like to show that intervals given by $\{ [0,t] : t \in \mathbb{R}_+ \}$ satisfy that ...
1
vote
1answer
49 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
3
votes
3answers
71 views

Measure theory convention that $\infty \cdot 0 = 0$

In the preface of Terry Tao's notes on measure theory he states that in the extended real number setting we adopt the convention that $\infty \cdot 0 = 0 \cdot \infty = 0.$ He explains that it's a ...
2
votes
1answer
48 views

Is $\mathbb E(X|\mathcal G)$ an integral of $X$ with respect to some measure?

My instructor defined $\mathbb E(X|\mathcal G)$ in the usual way and mentioned that it can also be characterized as an integral of $X$ with respect to some measure. Similarly to $\mathbb E(X|A)=\int X ...
2
votes
1answer
29 views

Whether or not a certain function is measurable

Let $(X,\mathscr{A},\mu)$ be a measure space, let $f:X\rightarrow[0,\infty)$ be measurable, and let $u_n:X\rightarrow(0,\infty)$ be measurable for each $n\in\mathbb{N}$. I want to know if $\left( 1 + ...
0
votes
1answer
44 views

Show that it is at most countable

In a space of finite measure, show that a family of disjoint measurable sets with positive measure is at most countable. Could you give me some hints what I am supposed to do??
4
votes
2answers
34 views

Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
0
votes
1answer
42 views

What's the relationship between Borel set and set whose boundary is measure zero?

Is a set whose boundary is measure zero a Borel set? Does any given Borel set has a measure zero boundary? I want to give my ideas first: If $E \subseteq R^n $ is some set whose boundary has ...
0
votes
0answers
9 views

Jordan-measurability of balls

I'd like to show that balls are Jordan-measurable in $\mathbb{R}^n$ with the simplest possible argument. For now, what I have in mind is to say that the boundary of the ball is the union of $2^n$ ...
0
votes
1answer
28 views

Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
1
vote
1answer
84 views

Show that the measure is Lebesque

I want to show that a measure is the same as the Lebesque measure. How can I do that?? What properties does this measure has to satisfy so that it is Lebesque??
2
votes
1answer
26 views

Continuous Measures: Range

Let $\Omega$ be a sigma-finite measure space with no atoms. (Reminder: A subset $A\in\Sigma$ is an atom if $\mu(E)<\mu(A)$ implies $\mu(E)=0$ for all $E\subseteq A$.) Then the measure attains ...
0
votes
2answers
33 views

Help with Rudin's Riesz Representation Theorem

I am having trouble with the beginning of the proof of the Riesz Representation Theorem from Rudin. I will be assuming (for now, I will correct this later) that the notation is familiar to everyone. ...
1
vote
3answers
41 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
3
votes
0answers
35 views

Is there a version of L'Hopital Theorem in Measure Theory?

I was checking the proof of the theorem (or one of the proofs) and the Mean Value Theorem is used which immediately says this proof cannot be modified (a priori) for a Measure version, however, is ...
0
votes
0answers
20 views

Measurability of a function iff its components are measurable

I'm trying to prove that a function $$f=(f_1,f_2): \Omega \to E\times F$$ is measurable, that is, $f:(\Omega, \mathcal{A})\to (E\times F, \mathcal{E}\otimes \mathcal{F})$ is measurable iff ...
1
vote
2answers
45 views

Show that the measure is equal to zero

Let $\mu$ be a Borel measure in $\mathbb{R}$ such that $\mu(I) \leq v^a(I)$ for each bounded interval $I$, where $a>1$. Show that $\mu=0$. ($v(R)$ is the volume of $R$) Do we maybe use the ...
0
votes
1answer
93 views

Measures: Atom Definitions

Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!) Consider a subset $A\in\Sigma$. Then according to the wikipedia article it is an atom ...
1
vote
0answers
21 views

Product of counting measure and the integral

Given the sigma algebra $P(\mathbb{N}^2)$(or $P(\mathbb{Z}^2)$ and counting measure $n$, I need to show that $n \times n$ is the counting measure for the aforementioned sigma algebra and compute the ...
-1
votes
0answers
59 views

Complex Measures: Lebesgue Decomposition

Disclaimer: This thread is related to: Singular Continuous Measures: Definition? Context Let $\Omega$ be a measure space with finite measure $\mu<\infty$. Consider a finite measure ...
0
votes
0answers
20 views

Existence of a A measurable function

Let A be sigma algebra having subsets of R only. We define a function from subset of A to R is said to be A measurable iff every borel set is pulled back to elements of A. Is there a sigma algebra B ...
1
vote
0answers
32 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
0
votes
1answer
12 views

Help with proof of Jensens inequality

I'm trying to prove Jensens inequality, where $f \in L ^1(\mu) $, $a<f<b $ and $\phi $ convex on $(a,b)$, but are stuck on the last part part of the proof. I define $t= \int _{\Omega } f d \mu ...
0
votes
2answers
54 views

Graph of measurable function has measure 0 in the product measure space

I have the following homework problem: Let $(X, M , \mu )$ be a $\sigma$-finite measure space. Show that the graph of any measurable function $f: X \rightarrow \mathbb{R}$ has measure 0 in the ...
2
votes
1answer
26 views

question on existence of open set

Let $U$ be a bounded open set in $\mathbb{R}^n$ and $A$ be an open subset of $U$. Fixed $\epsilon >0$. Does there exist an open set $B \subset U$ such that $B \cap \overline{U} \ne \emptyset$ and ...
1
vote
2answers
84 views

Please recommend a problem book with solutions on graduate level real analysis and measure theory

I'd like to find a problem book (with solutions) about graduate level real analysis (measure theory). That is, at the level of the books by Royden or Zygmund.
0
votes
0answers
4 views

Maximization of an integral based on Vittali covering

Let $f \in L^1(Y)$ and $f\geq 0$. Where $Y = [-1,1]^N$, suppose $\{B(y_k,\epsilon_k)\}_k$ forms a Vitalli covering of $Y$, satisfying \begin{equation}\begin{aligned} &(i) B(y_k,\epsilon_k) ...
2
votes
2answers
27 views

Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$

Let $f_{m,n}(x)$ be a sequence (dependent on $m$, $n$) of Lebesgue integrable functions on $\mathbb{R}$. Suppose that $f_{m,n}(x)\to 0$ as $m,n\to+\infty$, for almost $x\in\mathbb{R}$; in addition, ...
1
vote
1answer
35 views

necessary conditions of measure approximation theorem

Measure approximation theorem (I can't really remember its exact name) states that let $A$ be an algebra, $\mu$ a measure on $\sigma(A)$ and $\mu$ is $\sigma$-finite on $A$. Let $E\in \sigma(A)$ such ...
6
votes
0answers
120 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
3
votes
0answers
38 views

Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
2
votes
1answer
32 views

Essential range of a function

Let $A_f$ be the set of all averages $\frac{1}{\mu(E)}\intop_{E}\,f\,d\mu$ where $E$ is of positive measure. What is the relationship between $A_f$ and $\mathbb{R}_f$? Is $A_f$ always closed? Are ...
2
votes
0answers
37 views

question about showing integral is zero

Given a real valued function $f:X\rightarrow\mathbb{R}$ we have that $(X,\mathcal{F},\mu)$ is a measure space and $f$ is measurable. If $\mu(\{x\in X\;|\; f(x)\in B\})=\mu(\{x\in X\;|\; -f(x)\in B\})$ ...
2
votes
0answers
25 views

Equality of two $\sigma$-algebras on $\mathbb{R}/\mathbb{Z}$

I need help with this problem: Let $p > 1$ a integer, $X = \mathbb{R} / \mathbb{Z}$, $\mathcal{B}$ the borel $\sigma$-algebra of $X$ and $T \colon x \mapsto px \text{ mod } 1$ I think that ...
3
votes
1answer
213 views

Why is it enough to prove the sentence?

I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume. At the beginning of the proof there is the following sentence: It is enough to look at the ...
0
votes
0answers
34 views

About measurability for operator-valued functions

Being $E_1$ and $E_2$ Banach spaces, and working in a finite measure space, I have the following two definitions of measurability for a function $f:\Omega\to\mathcal{L}(E_1,E_2)$: $\bullet$ I say a ...
2
votes
1answer
51 views

A question on Lebesgue measure: Inequalities

Let $A_n$ be a decreasing sequence of Borel sets with finite but with measure $\geq \epsilon > 0$ for all $n$. Then there exists a compact set $B_n \subset A_n$ for all $n$ such that $\mu(A_n ...
1
vote
1answer
37 views

Sequence of functions divided by constants converge to zero

I am trying to show the following: If $\{ f_n \}$ is a sequence of a.e. real-valued measurable functions in X, and the measure $\mu(X) < \infty$, there exist positive constants $a_n$ such that ...
1
vote
1answer
22 views

Does integrating out a variable in a two-variable measurable function produce a measurable function?

This problem is not a mere consequence of Fubini’s Theorem, so I thought that it would be suitable for posting here on MSE. Let $ (X,\Sigma,\mu) $ and $ (Y,\text{T},\nu) $ denote $ \sigma $-finite ...
-2
votes
2answers
54 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
3
votes
1answer
36 views

Characterization of the Haar measure in terms of the integrals of characters

I was reading a paper and I think that they used the following theorem: Let $G$ compact group and $\mu$ a probability measure on $G$. If $$\hat{\mu}(\xi)= \int_G \overline{\xi(x)} d\mu(x) = ...
0
votes
1answer
41 views

Lebesgue Mean Value Theorem

Disclaimer: This proof is taken out from Rudin, Real and Complex Analysis. Let $\Omega$ be a finite measure space $\lambda(\Omega)<\infty$. Denote the mean value by: ...
1
vote
1answer
28 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
1
vote
1answer
35 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
3
votes
1answer
52 views

Is the following a semiring?

I have the following problem: Let $f: X' \rightarrow X$ be any map and $\mathcal{H} \subseteq \mathcal{P}(X)$ a semring. Is $f^{-1}(\mathcal{H})$ a semiring? Thanks for your help!
1
vote
0answers
33 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...
1
vote
0answers
34 views

Showing that set contains no intervals.

Hi, I'm trying to solve a problem: Here goes: First part is true because sets $B$, $B'$, and $E$ are countable and hence $F$ is countable so it's Lebesgue measure is $0$, thus it contains no ...
0
votes
1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...