Tagged Questions

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

learn more… | top users | synonyms (1)

2
votes
1answer
24 views

Borel measure and Riesz measure

Let $\mu$ be a Borel measure on $\mathbb{R}^n$ s.t. $\mu(K)< \infty$ for all compact $K$. Show that $\mu$ is the restriction of some Riesz measure on $\mathcal{B}$. I try to prove it using the ...
0
votes
0answers
34 views

Characterization of Lebesgue measure based on translation invariant

I am trying to solve a problem about characterization of Lebesgue measure. Let $(\mathbb{R}^n, \mathcal{B}, \mu)$ be a Borel measure space whose measure $u$ is translation invariant and exist a set ...
0
votes
3answers
80 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...
0
votes
1answer
12 views

a bounded function is converge in measure, then its limit is also converge

If a series function, ${f_n} \rightarrow f $ in measure $\mu$, and $|f_n| \leq M$, how to show that $|f| \leq M$? My instructor gave a hint as follows, but I do not believe the first inequality ...
1
vote
0answers
39 views

Proving that any measure is the sum of a semi-finite measure and a measure which takes either 0 or infinity.

I need help proving that for a measure space (X, A, u) that u can be written as the sum of a semi-finite measure and a measure that takes on the values 0 and infinity. The second measure, u_i I ...
0
votes
1answer
46 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
0
votes
1answer
19 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...
0
votes
1answer
31 views

Covering Class and Describing Outer MEasure for General Measures

I am uncertain if my description is correct, but I describe the measure in a piecewise type fashion. In general, $\mu_{\lambda}^*(A) = \infty$, if $A = X$ or $A$ uncountable. $\mu_{\lambda}^*(A) = ...
1
vote
1answer
53 views

Prove that $\mathcal B(\mathbb R)\times \mathcal B(\mathbb R)\subseteq \mathcal B (\mathbb R^2)$

I need to prove that $$\mathcal A(\mathcal B(\mathbb R)\times \mathcal B(\mathbb R))= \mathcal B (\mathbb R^2)$$ Where $\mathcal B$ is the generated Borel algebra and $\mathcal A$ is the generated ...
2
votes
0answers
66 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
0
votes
1answer
34 views

absolute continuity - Dirac measure with respect to gaussian measure [duplicate]

Let $a \in \mathbb{R}$ and Dirac measure $\delta_a (A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$, and let $\mu_1$ be the one-dimensional gaussian measure. Let $\mu$ and $\nu$ be two ...
0
votes
1answer
40 views

Name for LDC: Lebesgue?

Is there also a name associated to the Lebesgue dominated convergence theorem like Beppo-Levi or Fatou? Would Lebesgue be reasonable? Who did originally prove it?
0
votes
1answer
21 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
0
votes
0answers
57 views

integral with respect to Dirac measure

Let $\delta_a(A)$ be the Dirac measure, that is $\delta_a(A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$ and $\phi : \mathbb{R} \rightarrow \mathbb{R}$ a bounded Borel function. What does ...
0
votes
1answer
54 views

What does it mean m(dx), where m is Lebesgue measure?

Let a $\in \mathbb{R}$, $\phi_n : \mathbb{R} \rightarrow \mathbb{R}_+$, $\phi_n (x) : = \frac{n}{\sqrt{2 \pi}} e^{\frac{-n^2 x^2}{2}}$, $n \geq 1$ and let $\mu_n (d x) : = \phi_n (x - a) \lambda (d ...
1
vote
1answer
48 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
2
votes
1answer
32 views

Approximation theorem from measure theory

Let $a$ be an algebra, $\mu_0$ a pre-measure on it, and $\mu$ be a measure on the generated $\sigma$-algebra. Let $E \in \sigma (a) $, such that $\mu (E) <\infty $. Show that $\forall ...
2
votes
2answers
75 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
1
vote
1answer
28 views

Definition of a $\sigma$ - finite set

I know the definition of a $\sigma$-finite measure. But I found a problem in which it asks to show a particular set is $\sigma$ finite? But what is a $\sigma$ finite set? This is the problem I found. ...
4
votes
1answer
52 views

If $X_n \rightarrow X$ almost surely then $f(X_n) \rightarrow f(X)$ almost surely

Proof: If f is continuous and $X_n \rightarrow X$ almost surely, then $f(X_n) \rightarrow f(X)$ almost surely. Thats the only information I have. Does this only hold if the measure on the target ...
0
votes
1answer
24 views

$L^p$ spaces and converging sequence in this space

I have a question about $L^p$ space which I kan not solve it could u plz help me: let $(\Omega,A, \mu)$ be a measure space and let $1<p<\infty$.let $f_n$:$\Omega$ $\to$$\mathbb{C}$ be a ...
1
vote
1answer
36 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
0
votes
2answers
42 views

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Just learned Lebesgue outer measure from Royden's Real Analysis. Let me give my proof. First, let $A$ be the set of irrational numbers in [0,1]. So $A\subset [0,1]\Rightarrow m^*(A)\le ...
0
votes
1answer
33 views

Between bayesian and measure theoretic approaches

I was wondering how a bayesian statistician would approach the problem of defining a probability density function for a random variable. In a measure theoretic sense, If the distribution of the ...
1
vote
0answers
55 views

Is there a dense set of positive measure which does not contain any open set?

I want to construct $A\subseteq[0,1]$ with $m(A)>0$ and for every open subset $U$ of $[0,1]$, $0<m(A\cap U)<m(U)$ and $U\not\subseteq A$. I think these sets must have measure zero. ...
0
votes
1answer
36 views

which one is integrable acoording to f & g

Let f, g : R $\to$ R be integrable functions. Show which of the following functions are necessarily integrable: a)$f^2$ b)$f^{1/3}$ c)$f(x)\sin(x)$ d)$\arctan(f)$ e)$\sqrt{\mid ...
3
votes
1answer
48 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
45 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
44 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
11 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
85 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
46 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
60 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 ...