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

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0
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2answers
58 views

Supported function within $[a,b]$

Suppose $\phi_n : \mathbb{R} \to \mathbb{R}$ and $|\phi_n(x)| \le 1$ for all $x \in \mathbb{R}$. Also suppose that $\text{supp } \phi_n \subset [a,b]$, with $a,b \in \mathbb{R}$ s.t. $a > b$, ...
1
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2answers
20 views

Example that fail to be $\lambda$-measurable

I am looking at this definition: Given an outer measure $\lambda$ on $X$, we called a subset $A$ in $X$ $\lambda$-measurable if for every subset $B \subset X$, $$\lambda(B)=\lambda(B \cap A) + ...
1
vote
1answer
41 views

$L^p$ space as a vector space

I know that the set $L^p(\Omega) = \{f:\Omega \rightarrow R : (\int_\Omega |f|^p)^{1/p}<\infty\}$, where $\Omega \subset R^n$, is a normed vector space. However, if I let $f$ be an extended ...
5
votes
0answers
29 views

Showing that $f_n\to f$ a.e. $\implies f_n\to f$ almost uniformly where $|f_n|\leq g\in L_1$ [duplicate]

The following problem is from Carothers' Real Analysis: Suppose $f_n$ is a measurable sequence of functions such that $|f_n|\leq g\in L_1$ for all $n$. Prove that $f_n\to f$ almost everywhere ...
1
vote
1answer
20 views

Relation between vague convergence and weak convergence

This is the Portemanteau Theorem. And this is its corollary. I tried to prove that (i) implies (ii) in this corollary using the Portemanteau Theorem above. But I have kept failed... What is so ...
1
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0answers
50 views

why( in measure theory ) zero multiply infinite is zero?

why( in measure theory ) zero multiply infinite is zero? I think because the area of line in the plane is zero then zero multiply infinite is zero,is it true?
2
votes
3answers
28 views

Borel sigma-algebra over [0,1]

I just started studying this, so forgive me if I get something wrong. I have been given the following definition of a Borel $\sigma$-algebra over $\Omega=[0,1]$: It is the smallest $\sigma$-algebra ...
1
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0answers
73 views

Representation of symmetric functions

Let $n \in \mathbb{N}$. Show that every symmetric function $f\colon E^n \rightarrow \mathbb{R}$ can be written in the form $f(x) = g\Bigl(\frac{1}{n}\sum_{i=1}^n \delta_{x_i} \Bigr)$, where $g$ has ...
1
vote
1answer
17 views

Convergence in distributiom

I need to show that for arbitrary random variables $X_n$, there exist a sequence of positive constants $a_n$ such that $a_nX_n\overset{D}\rightarrow 0$. Thus, I need to show that $\lim_{n\rightarrow ...
0
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0answers
23 views

Show $\mathscr B(\mathbb R) \otimes \mathscr B(\mathbb R) = \mathscr B(\mathbb R^2)$

How do I show $\mathscr B(\mathbb R) \otimes \mathscr B(\mathbb R) = \mathscr B(\mathbb R^2)$ where $\mathscr B$ is Borel Algebra. I don't quite sure how to prove this relation, do I show that they ...
2
votes
0answers
50 views

What do we call $\mathfrak{M}$?

I am starting to learn some measure theory, and I was wondering if there is a name for $\mathfrak{M}$. I have the definition: A collection $\mathfrak M$ of subsets of a set $X$ is said to be a ...
2
votes
1answer
23 views

algebra generated by finite set

Is algebra generated by a finite set $A$ same a the $\sigma$-algebra generated by the same set $A$? For example: $X=\{1,2,3,4\}$, $A=\{\{1,2\},\{ 2,3\},\{ 4\} \}$, what is the algebra generated by ...
2
votes
1answer
38 views

A basic question regarding Lebesgue's density theorem

Here is the question from Pugh's Real Mathematical Analysis: My answer to $b)$ is that for a closed square, points on corner has density $1/4$, while on the sides the density is $1/2$. But how to ...
1
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1answer
22 views

Proving certain function is a measure

Let $(X,\Sigma)$ a measurable space. Let$\mu:\Sigma \to \mathbb R_{\geq 0}$ be the set function that satisfies: (a)$A,B \in \Sigma$ and $A \cap B=\emptyset \implies \mu(A \cup B)=\mu(A)+\mu(B)$ (b) ...
0
votes
1answer
19 views

Cavalieri's Principle in measure theory

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable. Here, it uses "almost every", so what is an example where ...
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0answers
22 views

Lebesgue Measurable By Alternative definition of Measure

Prove that any compact set $K$ in $R^{n}$ is Lebesgue measurable and $m(K) < \infty$ Actually the proof of this is given in Stein and Shakarchi's book on Real Analysis (Page 38, Property 4) where ...
1
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0answers
44 views

If 2-dimensional Lebesgue measure is one then $S_x$ has 1-dimensional measure of one

I am working on a problem$^{(1)}$ like this: Suppose $A$ is a Lebesgue measurable subset of [0, 1]$^2$ with 2-dimensional Lebesgue measure $m_2 (A) = 1$. Show that for almost every $x \in [0, 1]$ ...
1
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1answer
39 views

Prove $\cos(n_k x)$ does not converge pointwise on $[0,2\pi]$

There are strictly increasing sequence $n_1<n_2<\dots<n_k<\dots$ which are positive integers. I want to prove on the domain $[0,2\pi]$ where $\cos(n_kx)$ converges does not coincide with ...
4
votes
1answer
124 views

A quite strange question about measure

this strange question just occurred to me, that say $E\subset[0,1]$ and $mE=1$, does it imply the closure $\overline E=[0,1]$? Or is there a closed proper set in $[0,1]$ that has measure $1$? In that ...
0
votes
0answers
25 views

Yet another proof of uniquness of Haar measure

I'm trying to prove the uniqueness of Haar measure in my way. Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. ...
1
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0answers
34 views

Prove that $f(x)=e^{-1 /\Vert x \Vert^\alpha}$ is measurable function.

Let $\Omega$ be a bounded domain in $\mathbb{R^N}$ and $f(x)=e^{-1 / \Vert x \Vert^\alpha}$. Prove that $f(x)$ is a measurable function. I wonder if I have to add more condition to $f(x)$. For ...
4
votes
1answer
70 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
2
votes
1answer
41 views

Is this statement about measurable function true? [duplicate]

If $f$ is a measurable function on $[0,1]$, then there exists a measurable set $D\subset [0,1]$ such that $mD>0.9$, and a continuous function $g:[0,1]\to R$ such that $f=g$ on $D$. My question is ...
0
votes
3answers
47 views

$\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, $k < l$. [closed]

How do I show that $\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, with $k < l$?
1
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1answer
33 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
2
votes
1answer
40 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
1
vote
0answers
41 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
1
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0answers
23 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
1
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0answers
23 views

continuity at a point in a cadlag process

I am reading a proof that uses the fact: Let $(X_t)_{t \geq 0}$ be a cadlag process. We know that $X(\omega)$ has at most countably many discontinuities, for each $\omega \in \Omega$. It is then ...
2
votes
0answers
17 views

For $f\in L^1_{loc} (\Omega)$, $f=0$ almost everywhere in $\Omega$ provided $\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$

I need to show that $f=0$ almost everywhere in $\Omega$ provided $$\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$$ Here is how I have decided to proceed. Suppose there ...
1
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1answer
21 views

A problem about Markov kernel and the Monotone class theorem

Let $f: A\times B \times C \to R $ be a measurabled function, which is also bounded. And $p:B \times \sigma(C)\to R$ be a Markov kernel. Prove $g(x,y)=\int f(x,y,z)dp_y (z)$ is measurable with ...
1
vote
1answer
22 views

A relation between convergence in measure and pointwise convergence

Let $\{f_n\}$ be a sequence of measurable functions on $R$ with Lesbegue measure and $f$ be a measurable function. I have to show that $\{f_n\}$ converges to $f$ in measure if and only if any ...
1
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0answers
34 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
1
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0answers
27 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
1
vote
1answer
20 views

Proof of property of Lebesgue measurable subset of $\mathbb{R} $ with non-zero measure

I'm having trouble getting my head around how to prove the following: Let $A \subset \mathbb{R}$ with $m(A)>0$. Then $\forall \epsilon \in (0,1)\ \exists$ an open interval $I$ such that $m(A\cap ...
0
votes
1answer
28 views

$f_n$ converge to $f$ almost everywhere (measure theory)

If $\mu(\{w \in A: |f_n(w) - f(w)| > \epsilon\,\, \text{for infinitely many many $n$}\}) = 0$ for each $\epsilon > 0$, then $f_n \rightarrow f$ $\mu$-a.e. I can see how to solve this problem ...
0
votes
1answer
40 views

prove the equivalence between a null set and a limit

I'm asked to prove that for any non-negative, measurable and integrable function $f$ on $[0,1]$, we have $\lim\limits_{a\to 0}\int_{0}^{a}fdx=0$. I want to use the theorem that for null set E, such a ...
0
votes
2answers
34 views

Almost everywhere convergence for integral

I have a question about the following proof (from Rudin's RCA). And, here are the Theorems 1.27 and 1.34 What I do not understand about Theorem 1.38 is the conclusion that the series $f(x)$ ...
0
votes
1answer
31 views

Jordan outer content of rationals in [0, 1]

How to prove Jordan outer content of rationals in $[0,1]$ is 1 just by definition of Jordan outer content? I mean how to prove this without using "Jordan content of a set is equal to that of its ...
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votes
0answers
45 views

If $f$ and $g$ are measurable, is $\{x:f(x)=g(x)\}$ measurable?

My question is if $g$ and $f$ are measurable then $\{f=g\}$ is also measurable. Here is my answer, but I need you to tell me if it is true. Let $\{x\in \mathbb{R}:f(x)˃a\}$ and $\{x\in ...
5
votes
3answers
113 views

Why is the relation “$f=g $ almost everywhere” transitive?

In Rudin's Real and Complex Analysis, it says on pg 27 that If $\mu$ be a measure, define $f\sim g$ iff $\mu(\{x|f(x)≠g(x)\})=0$, where $f,g$ are measurable functions from $X$ to a topological space ...
2
votes
2answers
40 views

Comparing limits of integrals

If $$f_n:X\rightarrow [0,\infty]$$ is a sequence of measurable functions and we know that $$\lim_{n\rightarrow \infty }\int_X f_n \,d\mu=0,\qquad \qquad \tag{$\star$}$$ then can we conclude that ...
-2
votes
1answer
38 views

The Pólya urn model describes a martingale

Suppose an urn contains one blue and one red ball and that we perform the following random experiment: In each round $n\in\mathbb{N}$ we randomly draw a ball If the drawn ball is blue, we replace it ...
1
vote
0answers
32 views

Proof Borel Sigma Algebra

Let $I \equiv \lbrace [- \infty, a[ : a\in \mathbb{R}\rbrace$. Is $\sigma(I)$ Borel's sigma algebra on $\mathbb{R}$? I'm having difficulties proving these statement. I suppose it's not the Borel's ...
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votes
1answer
46 views

Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? [closed]

As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?
2
votes
1answer
18 views

Marginals of (not necessarily finite) measures

Consider a product of two measurable spaces, $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, and a (not necessarily finite) measure, $\varrho$ on the product space $(X \times Y, \mathcal{A} \otimes ...
3
votes
2answers
45 views

Why is it true that every set in $\mathbb R^n$ can be covered by a countable union of open balls?

Why is it true that every set in $\mathbb R^n(n\geq 2$ can be covered by a countable union of open balls? This is the fact we are using in the definition of measure theory where outer measure of a ...
5
votes
1answer
93 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
0
votes
0answers
47 views

Simple proof check---Exercise from Tao's real analysis notes, Jordan measure

It is a rather simple question, I just want to make sure if my way of doing it works. So the exercise is to proof the monotonicity of Jordon measure (http://en.wikipedia.org/wiki/Jordan_measure). If ...
0
votes
1answer
19 views

Can ratios similar to those related to the surface area of a circle and sphere be applied to determine properties of a 3-sphere?

Applying the strategy of describing the surface area of a circle as a product of the ratio for the surface area of a triangle, reveals a consistency that also applies to the surface area of a cone. ...