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

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2
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2answers
123 views

Uniform convergence and pointwise convergence, understanding

Pointwise convergence $\ \ \Leftrightarrow \ \ \{f_n(x)\}_{1}^{\infty}$ converges to $f(x)$ for all $x\in X$ Uniform convergence $ \ \ \Leftrightarrow \ \ \forall \epsilon > 0 \exists N\in\mathbb{...
1
vote
2answers
78 views

Can Any one give me some hints? [closed]

Can Any one give me some hints? Please dont give me the answer i still want to left some room for myself to think .May anyone give me some advice so that i can solve the question Question: Let $K$ be ...
1
vote
1answer
31 views

Showing that the function is measurable.

I have that $\mu$ is a $\sigma$-finite Borel measure on $\mathbb{R}$, and $E \in \mathcal{B}$. I need to show that the function $\mu(E-x)$ is Borel-measurable. That is, the inverse image of any open ...
1
vote
1answer
36 views

Integration of Nonnegative function, Folland Real Analysis using MCT

Suppose $\{f_n\}_{n=1}^{\infty}\subset L^{+}$, $\lim_{n\rightarrow \infty} = f$ pointwise, and $\int f d\mu = \lim_{n\rightarrow \infty}\int f_n d\mu < \infty$. Then $\int_{E}f d\mu = \lim_{n\...
2
votes
1answer
44 views

Total variation distance of two random vectors whose components are independent

Let $X^n=\left(X_1,\ldots,X_n \right)$ and $Y^n=\left(Y_1,\ldots,Y_n \right)$ be such that all $X_i$'s are independent and all $Y_i$'s are independent. I am trying to prove the following: $$ d_{TV} \...
0
votes
1answer
37 views

Integration of Nonnegative funtion, Real Analysis Chapter 2 problem 16

If $f\in L^{+}$ and $\int f < \infty$, for every $\epsilon > 0$ there exists $E\in M$ such that $\mu(E) < \infty$ and $\int_{E} > \left(\int f\right) - \epsilon$ Proof: Let $\epsilon > ...
2
votes
1answer
71 views

Lebesgue monotone convergence theorem

I have a doubt regarding the Lebesgue monotone convergence theorem. The version that I know is the following from Wikipedia, requiring, in particular, $\{f_k(x)\}$ monotone increasing and $f_k(x)\...
1
vote
1answer
26 views

An strange multiplication of measurable functions

Let $I$ be an uncountable (directed) set. Let $\{f_i\}_{i\in I}$ and $\{g_i\}_{i\in I}$ be two set of complex valued measurable functions on a measurable space $\Omega$. For every $t\in \Omega$, we ...
1
vote
2answers
31 views

Showing that a measure is finite

I try to solve the following task and I'm not sure, if I did it correct: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f:\Omega\rightarrow]0,\infty[$ is a measurable function so that $f$ ...
1
vote
1answer
20 views

Prove sum is commutative with measure theory result

Given a measured space $(X,\mathcal{A},\mu)$, consider the following function $f:X\rightarrow \mathbb{R}^+$ defined by $$ f(x)=\sum_{n=0}^\infty f_n(x), $$ where $f_n(x)$ are positive measurable ...
2
votes
1answer
21 views

Determining if a set is measurable by upper and lower sets

I have the following question regarding Lebesgue measure: If $A,B$ are measurable sets and I have $m(A\setminus E)=0$ and $m(E\setminus B)=0$, is it enough to determine that $E$ is measurable? We do ...
0
votes
1answer
67 views

Use $\int_S f^{\pm} d\mu = \lim \int_S f_n^{\pm} d\mu$ to prove Scheffe's Lemma

Let $(S, \Sigma, \mu)$ be a measure/probability space. Scheffe's Lemma Part (ii): Suppose $\{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu)$ and $\lim_{n \to \infty} f_n(s) = f(s) \...
0
votes
1answer
43 views

If $\int_a^x f(t)\, dt$ is differentiable, is its derivative integrable?

Let $f$ be a real-valued Lebesgue integrable function on $[a,b]$. If $F(x) = \int_a^x f(t)\, dt$ is differentiable on $[a,b]$, is $F'(x)$ (Lebesgue) integrable? I know there are examples of ...
2
votes
0answers
39 views

Convergence in measure implies convergence in Frechet's distance

Let $(X,S,\mu)$ be a finite measure space and let's define: $$d(f,g):=\int \frac{|f-g|}{1+|f-g|}\,d\mu\;\;\forall f,g\in M(X,S)$$ I want to prove that if $f_n\ \xrightarrow[\mu]{} f\Rightarrow d(...
3
votes
1answer
47 views

Proving the continuity of a function with respect to a measure

Let $\mathcal{M}([0,1])$ be the space of all real finite measures on $[0,1]$, with norm $\|\mu\|=|\mu|([0,1])$ and consider the function $$u(y)=\int_{[0,1]}\min\{x,y\}\mu(dx)$$ for $y\in[0,1]$. I ...
0
votes
0answers
66 views

Modes of convergence in Measure Theory

I am currently trying to learn about modes of convergence in Measure theory and am struggling to understand the difference between definitions of uniform and pointwise. I have either proofs or ...
2
votes
1answer
64 views

A Haar measure is a left invariant Borel measure which is not identically zero

A Haar measure is a Borel measure $\mu$ in a locally compact topological group $X$, such that $\mu$(U)>0 for every non empty Borel open set $U$, and $\mu(xE)$=$\mu(E)$ for every Borel set $E$. A ...
0
votes
1answer
20 views

Uniformly convergent subsequence example seems trivial is it correct?

Given a sequence of functions $f_n$ all continuous. If asked if there exists a subsequence $(f_{n_k})_{k \in \mathbb{N}}$ and continuous function $f$ in which $f_{n_k} \to f$ as $k \to \infty$, can I ...
0
votes
1answer
21 views

Normalization constant for $f(x) = \exp(-x^\alpha)$

If $f$ is a density with respect to Lebesgue measure given by $k e^{-x^\alpha}$, $x > 0$, then what constant $k$ makes $f \cdot m$ a probability measure? I need to compute $\int_0^\infty \exp({-x^...
2
votes
1answer
62 views

An inequality of integrals

Let $f \in L^{2}(\mathbb{R})$ be continuously differentiable on $\mathbb{R}$. I am trying to show the following: $( \int |f|^{2} dx)^{2} \leq 4 ( \int |xf(x)|^{2} dx) ( \int |f'|^{2} dx))$. My first ...
0
votes
1answer
51 views

Proving that $f'$ is measurable on $\mathbb R$ if$f$ is differentiable on $\mathbb R$

Since $f$ is differentiable on $\mathbb R$ it is then continuous on $\mathbb R$,making $f$ measurable. (this we proved in class-that continuous functions are measurable). I tried to use this to prove ...
0
votes
1answer
64 views

About Lebesgue-Radon-Nikodym Theorem

Lebesgue-Radon-Nikodym Theorem shows that: If $\mathbb{M}$ is a $\sigma$-algbra on the set X. $\mu,\lambda$ are a $\sigma$-finite positive measure and $\sigma$-finite signed measure on $\mathbb{M}$ ...
0
votes
2answers
12 views

Proving equivalence of operators imply equivalence of measures

Let $A:L^2([0,1],\mu)\to L^2([0,1],\nu)$ an unitary operator. Prove that $$d\mu=\rho(x) d\nu$$ for some $L^1(\mu) \ni \rho(x) >0 (\mu\text{ a.e})$ I thought maybe saying $$\int_{[0,1]}|f(x)|^2d\...
3
votes
1answer
105 views

Simple proof that a Lebesgue-measurable and additive function is linear

Let $(\Bbb R, \mathcal A_{\Bbb R}^*,\overline{\lambda})$ be the complete lebesgue-measure space. Let $f:\Bbb R\to \Bbb R$ be an additive function and also Lebesgue-measurable: $$\text{I want to prove ...
1
vote
1answer
56 views

sum of sigma finite measure

I want to demonstrate: Finite sum of sigma-finite measures is a sigma finite measure. I try this: Suppose that are m sigma measures in a space ($\mathcal{X}$,$\mathcal{A}$), denoted by $\mu_k$ with $...
1
vote
1answer
65 views

How to show convolution is smooth i.e. in $C^\infty$

$\newcommand{\Rn}{\mathbb{R}^N}$ $\newcommand{\dxi}[2]{\frac{\partial #1}{\partial x_{#2 } } } $ $\newcommand{\conv}[2]{#1 \star #2}$ $\newcommand{\ball}{B(0, {1 \over n})}$ I would like to show the ...
0
votes
1answer
31 views

Riemann sums for a Lebesgue integrable density function

Let $f$ be a non-negative measurable function whose support is $[0,1]$ and it's integrable. In other words, it's a density function of a continuous random variable which is $[0,1]$-valued. Then is it ...
4
votes
1answer
27 views

Prove that bounded $p$-norms and convergence a.e implies convergence in $L^1$

Suppose $\mu\left(X\right) < \infty$, $f_n \rightarrow f$ a.e and $p > 1$ is such that for some constant $C>0$, we have $$ \|f_n\|_p \leq C,\ \ \text{for each} \ n $$ Prove that $f_n \...
0
votes
0answers
39 views

Property of a Lebesgue measurable function

Let $\big(\Bbb R,\mathcal A_{\Bbb R}^*,\overline{\lambda}\big)$ be a lebesgue (complete) measure space and let $f:\Bbb R\to \Bbb R$ be a lebesgue measurable function. $$\text{I want to guarantee ...
3
votes
2answers
174 views

Sigma algebra - motivation in measure theory

Taken from the Motivation section of sigma-algebra article: A measure on $X$ is a function that assigns a non-negative real number to subsets of $X$; this can be thought of as making precise a ...
1
vote
1answer
23 views

Approximating measurable sets of infinite measure by open sets

Following a question posted here: Approximating measures by open sets and compact sets. I wanted to ask, if I am given a measurable set $E\subseteq \mathbb{R}$ s.t. $m(E)=\infty$, then how can I find ...
-3
votes
5answers
72 views

Difference between $A+B$ and $A \cup B$

Basic set theory: $A$ and $B$ are two sets. I assume that $A+B$ isn't the same as $A\cup B$. I know what $A\cup B$ is but what is $A+B$? The context: I need to show that if A and B are open, then A+B ...
1
vote
0answers
42 views

tricks to tell whether an integral function is absolute convergent.

With measure theory I am experimenting with Tonelli's theorem and it is going just fine, but I always get stuck at proving absolute convergence of integrals. Examples are: $$\int_{0}^{\infty} \int_{...
2
votes
0answers
38 views

How to use dominated convergence on $\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x) $?

I find it hard to find an appropriate dominating function for the integral $$I:=\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x), \ t > 0 $$ ...
1
vote
1answer
98 views

Good book for learing sigma algebra?

I am beginner in probability theory. In order to make a better understanding of Borel Sets, Measurable Space and Random Variable, I need to learn about algebra and sigma algebra, Can anyone please ...
-1
votes
1answer
71 views

Are all sets in sigma-algebra measurable?

In the Wikipedia article it says: the collection of those subsets for which a given measure is defined is necessarily a $\sigma$-algebra. Fine, but is the opposite true? Do we know for sure ...
1
vote
1answer
28 views

Measure of non-compactness

Can someone give me some simple examples of measure of non-compactness of sets in Banach spaces or metric spaces, which are easy to understand.
0
votes
1answer
26 views

Approximating continuous function vanishing at infinity on product space by product of functions

I am not sure if the question I am asking is proper enough: Is there any way to approximate any arbitrary function $f(x,y) \in C_{0}(\Omega $ X $ S)$ in the uniform norm by linear combinations of ...
0
votes
1answer
17 views

Proving finite additivity of measure of intervals without using riemann

I wish to show the following: Let $I$ be any finite interval (i.e. Bounded) of any type (open, closed, half open). 1) If $I_1,\dotso, I_n$ are also intervals, pairwise disjoint such that $$I=\bigcup_{...
1
vote
0answers
16 views

Does $n$-dimensional convolution integral always exist

If $X_1,X_2,\ldots,X_n$ are independent random variables with densities $f_1,f_2,\ldots,f_n$, then we know that the density of their sum $X_1+\cdots+X_n$ is given by $$ g(z)=\idotsint f_1 \left(x_1\...
2
votes
0answers
69 views

When can we restrict/condition a probability measure to a subset of zero measure?

If $\mu(C)>0$, $$\mu_C(A)=\frac{\mu(A\cap C)}{\mu(C)}$$ is well-defined. If $\mu(C)=0$ things get hairy. If $C=\{x\}$ (single point) then $\mu_C=\delta_x$. If $C=\bigcup_{i=1}^nx_i$ (a finite ...
1
vote
0answers
68 views

H. Steinhaus Theorem simple proof

Let $A,B\in\mathcal A_{\Bbb R}^*$ be given with $\overline{\lambda}(A)<\infty$ and $\overline{\lambda}(B)<\infty$. Let $\overline{\lambda}(A)>0$ I want to prove that: $$\exists\ \delta>0\;\...
3
votes
1answer
66 views

Showing the integral of a function is finite almost everywhere

Suppose$\ E \subset \mathbb R$ is closed. Let$\ d(y) = \inf \{|x-y| : x \in E \} $ and let $\ M(x) = \int_0^1\frac{d^a(y)}{|x-y|^{(1+a)}} dy $ , for some arbitrary constant $a$. Show that $\ M(x)$ is ...
1
vote
1answer
45 views

Integration of nonnegative funtion, Folland Real Analysis

Suppose $f$ is a nonnegative measurable function on a measure space $(X,M,\mu)$ satisfying $\int f d\mu < \infty$. Show that for every $\epsilon > 0 $ there exists a $\delta > 0 $ such that ...
0
votes
0answers
100 views

If $f_n \to f$ in measure and $|f_n| \leq g \in L^p$ for all $n$, then $||f_n -f||_p \to 0$

Suppose $1 \leq p \leq \infty$ If $||f_n - f||_p \to 0$ then $f_n \to f $ in measure, and hence some subsequence converges to $f$ a.e. On the other hand, if $f_n \to f$ in measure and $|f_n| \leq g \...
1
vote
1answer
45 views

The proof of the weakly sequential completeness of $L^1$ when the measure is $\sigma-$finite

In this old post, a sketch of the proof was given. My question is about some details in the proof(which are not given in the books). Let me simply copy what t.b. wrote in the solution: First of all,...
2
votes
1answer
57 views

Weak convergence $f_n \rightharpoonup f$ in $L^2([0, 1])$ as $n \to \infty$?

Assume that $\|f_n\|_{L^2([0, 1])} \le 1$ and $f_n \to f$ in measure as $n \to \infty$. How do I see that $f_n \rightharpoonup f$ in $L^2([0, 1])$ as $n \to \infty$?
0
votes
1answer
82 views

Showing a set has Jordan content

The question is: Let $a<b$ and $f\colon [a,b]\rightarrow[0,\infty)$ be continuous. Let $D = \{(x,y)\in\mathbb{R}^2\mid x\in[a,b],\: y\in[0,f(x)]\}$. Show that D has content and $$ \mu(D) = \...
0
votes
1answer
53 views

why is the outer measure of the unit interval in $R^2$ equal to zero

I was reading Terry Tao's second volume on Analysis and he makes the following remark on page 586. I was hoping that someone could explain why this is the case since he does not give a proof. "The ...
0
votes
0answers
69 views

Question about product measure (from Folland)

Folland chapter 2, question 51: I did the part (a), but the (b) is a little confusing for me. (b) Let $(X,M,\mu)$ and $(Y, N,\nu)$ be measure spaces, not necessarily $\sigma$-finite. Let $f\colon X \...