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

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1answer
56 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
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1answer
31 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
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1answer
34 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
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1answer
71 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
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0answers
79 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
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1answer
61 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 ...
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1answer
47 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?
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1answer
23 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, ...
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0answers
83 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
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1answer
69 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
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1answer
59 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
56 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 ...
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4answers
145 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
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1answer
48 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
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1answer
58 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
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1answer
27 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 ...
0
votes
2answers
172 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
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1answer
71 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 ...
0
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1answer
39 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
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1answer
241 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
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2answers
65 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
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1answer
29 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 ...
2
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1answer
91 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
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3answers
83 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
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1answer
55 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
31 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 + ...
1
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1answer
67 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
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2answers
59 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
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1answer
85 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 ...
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0answers
21 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
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1answer
44 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
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1answer
111 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
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1answer
32 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
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2answers
71 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
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3answers
50 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
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0answers
39 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 ...
1
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2answers
52 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
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1answer
139 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 ...
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0answers
40 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 ...
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1answer
88 views

Borel Measures: Lebesgue Decomposition

Disclaimer Please, if you don't like self-answers just avoid this thread. (For more details see: Answer own Question?) Context Given a measure space $\Omega$ with sigma-finite measure $\lambda$. ...
1
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0answers
29 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 ...
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0answers
46 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
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1answer
32 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 ...
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2answers
121 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
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1answer
33 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 ...
2
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2answers
209 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.
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0answers
10 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
39 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
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1answer
40 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
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0answers
128 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 $m(X)=1$. ...