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

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0
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2answers
22 views

Does a.e. convergence imply the boundness in $L^1$?

Let $f_n : I = (0, 1) \to \mathbb{R}$ be a sequence of functions. If $$f_n \to 0 \;\; a.e$$ does it imply that $$f_n \;\; \text{is bounded in} \;\; L^1(I)?$$ Why yes/not? Thank you!
2
votes
1answer
48 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
1
vote
1answer
34 views

What does it mean that a sequence of functions is bounded in $L^1(I)$?

Let $I = (0, 1)$ and $f_n : I \to \mathbb{R}$ a sequence of functions. What does it mean that $f_n$ is bounded in $L^1(I)$? Does it mean that $$\exists c>0 \;\; \text{such that} \;\; \|f_n\|_1 ...
0
votes
0answers
12 views

Prove for measures $\mu $and $\nu$ $\nu \perp \mu$ iff $|\nu| \perp \mu$ iff $\nu^+ \perp \mu$ and $\nu^- \perp \mu$

Where $\perp$ means mutually singular. I have a question, as $\nu$ is clearly a signed measure do we assume that $\mu$ is signed or just positive? It follows from $\nu\perp\mu$ with the set in ...
0
votes
0answers
16 views

Algebra and $\sigma$-algebra

We consider 3 intervals $A_1$, $A_2$ and $A_3$, which are defined as $$ A_1=\left(-\infty,0\right], ~A_2=\left(0,\frac{1}{2}\right], ~A_3=\left(\frac{1}{2},+\infty\right). $$ We then form the ...
1
vote
0answers
27 views

A set of the second category has a positive measure?

A subset of a topological space $X$ is called nowhere dense in $X$ if the interior of its closure is empty. A subset of a topological space $X$ is called the first category (or meagre) in $X$ if it ...
0
votes
1answer
24 views

Doubt on Caratheodory's extension theorem

This doubt is on the Caratheodory's extension in Billingsley. The main theorem says that a countably additive probability measure $P$ on a algebra can be extended to a countability additive ...
1
vote
0answers
34 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
0
votes
0answers
11 views

Regular measure on Borel sets

I am trying to do the following problem: Let $\mu$ be a measure defined on the Borel sets of $\mathbb R^n$ such that $\mu$ takes finite values on the compact sets. Let $\mathcal H$ be the class of ...
0
votes
1answer
25 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...
0
votes
1answer
69 views

Of which sets does the $\sigma$-algebra generated by the first $n$ singletons of $\mathbb{N}$ consist?

Let $n \in \mathbb{N}$ and $\mathcal{E}_n := \{\{1\},\{2\},\dots,\{n\}\}$. The $\sigma$-algebra which is generated by $\mathcal{E}_n$ is defined as follows: $$\sigma(\mathcal{E}_n) := \bigcap ...
2
votes
2answers
31 views

Why is the zero extension of an $L^p$ function in $L^p$?

Let $u \in L^p(0,1)$. Define $\tilde u:(0,\infty) \to \mathbb{R}$ as the function which equals $u$ on $(0,1)$ and $\tilde u =0$ on $(1,\infty)$. I cannot figure out why this function is measurable. ...
1
vote
0answers
16 views

Question on proof of disintegration of measures

In a probabilistic setting: Let $\mu$ be a measure on the product space $S=S_1\times S_2$, both standard Borel, $\mu_1, \mu_2$ the marginal measures. Then there exists a Markov kernel $k$ such ...
2
votes
1answer
45 views

Showing $E[X_{n+1}|X_1,…,X_n] = a_0+\Sigma_{k=1}^n a_kX_k$

$X_1,...,X_n,X_{n+1}$ are jointly distributed with a Gaussian distribution. We let $X^* = E[X_{n+1}|X_1,...,X_n]$. Show that there exists constants $a_1,...,a_n,a_{n+1}$ such that $X^* = ...
1
vote
0answers
49 views

About an idea in proving Riesz representation theorem for continuous function of compact support.

I tried to prove Riesz representation theorem for continuous linear functional defined on the continuous functions of compact support of a topological locally compact Hausdorff space in this way: 1) ...
1
vote
1answer
45 views

If iterated integral is zero then function is zero

We are in Measure & Integration class and were assigned this problem from a chapter on Product Measure & Fubini Theorem: Let $f$ be a real-valued function, integrable with regards to ...
4
votes
1answer
25 views

On showing that if $f_n \to f, g_n \to g$ in $L^p$ then $max(f_n, g_n) \to max(f, g)$ in $L^p$

Let $(f_n)$ and $(g_n)$ be two sequences in $L^p(\Omega)$ with $1 \leq p < \infty$ such that $f_n \to f$ in $L^p(\Omega)$ and $g_n \to g$ in $L^p(\Omega)$. Let $h_n = max(f_n, g_n)$ and $h = max(f, ...
0
votes
3answers
48 views

Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue measure

Let $ f : \mathbb R \rightarrow \mathbb R$ be a bounded Lebesgue measurable function such that $\int_a^b f =0$ for all real $a,b.$ Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of ...
0
votes
0answers
13 views

Gaussian Measure for Random Matrix

I am doing physics and do not have enough mathematical background. so the question may be trivial, I apologize for that. any help would be highly appreciated. my question is: How does Random Matrix ...
1
vote
1answer
45 views

Basic Properties of Integrals

I am doing a course on measure theory and we are studying integrals and there is a lemma which states: $\int f$ $d\mu$ exists and is greater than $-\infty$ if and only if $\int f^{-} d\mu$ $< ...
1
vote
0answers
21 views

There is a Borel measure $ \mu $ on $ [0,1] $ such that $\mu([0,x))=G(0)-G(x) \forall{x\in (0,1]}$

Let $C_c([0,1])=\{f \in C(X): supp (f)$ compact $\}$ and $G:[0,1]\to \mathbb{R}$ be a function monotone nondecreasing continuous and let $\bigwedge:C_c([0,1])\to \mathbb{R}$ define by ...
1
vote
1answer
16 views

Sigma-algebra: $\sigma(\mathcal{A})=\mathcal{A}$?

Let $\mathcal{A}$ be a $\sigma$-algebra; and let $\sigma(\mathcal{A})$ be also a $\sigma$-algebra generated by $\mathcal{A}$. Because $\sigma(\mathcal{A})$ is the smallest $\sigma$-algebra ...
0
votes
1answer
20 views

All Borel Sets are measurable

Given an outer measure $\mu^{*}:(X,d) \rightarrow [0, \infty]$, where $X$ is a metric space, will all Borel Sets be $\mu^{*}$-measurable only when $\mu^{*}$ is a metric outer measure? Because if ...
1
vote
1answer
43 views

Proving the existence of a certain Lebesgue-measurable set.

Let $ m $ be the Lebesgue measure on $ \mathbb{R} $ and $ f: \mathbb{R} \to [0,\infty) $ a Lebesgue-integrable function. Show that there exists a Lebesgue-measurable set $ E \subseteq [0,\infty) $ ...
3
votes
1answer
22 views

Convergence in some measure implies convergence in some other measure absolutely continuous to the first

Suppose that $f_n$ converges to $f$ in measure $\mu$ and $\nu$ is a different measure absolutely continuous with respect to $\mu$ (which of course means that both $\mu$ and $\nu$ are defined on the ...
1
vote
0answers
13 views

A probability problem on outer measure

Let $P$ be a probability measure on a algebra $\mathcal{F}_0$ and for every subset $A$ of $\Omega$ define $P^*(A)$ by $$P^*(A) = \inf \sum P(A_n)$$ where $A_n \in \mathcal{F}_0$ and $A \subset \cup ...
1
vote
4answers
56 views

Is every subset of a Borel set Lebesgue measurable?

Is it true that every subset of a Borel set is Lebesgue measurable? Why?
0
votes
1answer
21 views

Symmetric difference and Measurability

Let $(\mathcal{X},\mathcal{M},\mu)$ a measure space. Declare sets $E_1,E_2 \in \mathcal{M}$ to be equivalent if $\mu(E_1\Delta E_2)=0$ where $\Delta$ denotes the symmetric difference. Show that the ...
0
votes
0answers
26 views

Question on finite Borel measure on $\mathbb{R}$

Let $\mu$ a finite Borel measure on $\mathbb{R}$, and let $f_\mu(x) =\mu((-\infty,x])$. Show that $f_\mu$ is monotone increasing, $\mu((a,b])=f_{\mu}(b)-f_\mu(a) $ for all $a<b$, $f_\mu$ is right ...
1
vote
2answers
29 views

Measurable set, inner and outer measure

I am trying to show the following "If $E \subset \mathbb R^n$ is Lebesgue measurable then for every $A \subset E$, $$m(E)=m_i(A)+m_e(E \setminus A)$$ I got stuck trying to show this, I know that the ...
0
votes
0answers
9 views

Bourbaki's definition of the upper integral of an arbitrary non-negative function

Let $X$ be a locally compact Hausdorff space with a countable base. Let $\mathcal B$ be the $\sigma$-algebra generated by the set of open subsets of $X$. A measure $\mu$ defined on $\mathcal B$ is ...
0
votes
1answer
45 views

Question about working area of Vitali cover

$\text Carothers' Real Analysis$ defines his Vitali cover with no introduction which made me confused a lot. Here is the definition of a Vitali Cover: I'm not sure when does a set have its Vitali ...
2
votes
0answers
25 views

Enigma in applying Lebesgue dominated convergenege theorem

Let $p\in \mathbb{C} \lbrack z],~p=p\left( re^{it}\right) ,n=\deg p$ and we want to compute de limit $$ \lim_{r\rightarrow\infty}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{p^{\prime}\left( re^{it}\right) ...
3
votes
0answers
35 views

A strange Jensen's inequality for function of two variables

I am reading a paper where they use implicitly the following "Jensen's inequality which i find quite strange. Moreover i did not find this result in any textbook so, i would like an opinion before i ...
3
votes
1answer
46 views

Every Countable set has measure $0$

I came across the following lemma in some book: Every countable set has measure $0$ and the proof involved "breaking" down the countable set into a countable union of points and then proved that ...
5
votes
1answer
71 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
2
votes
1answer
32 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed (weak-* sense) set of probability measures. $m$ is a probability ...
9
votes
2answers
294 views

Lebesgue integration of simple functions

Define $f : [0,1] \to \Bbb R$ by $f(x) := 0$ if $x$ is rational, and $f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero digit in the decimal expansion of $x$. Show that ...
0
votes
0answers
9 views

If $S_M$ denotes the measure of a submanifold, then $\frac 1{r^{n-1}n\omega_n}\int_{\partial B_r}u(x)\;dS_{\partial B_r}(x)\to u(y)$ for $r\to 0$

Let $S_M$ denote the "surface measure" of a submanifold $m$ $B_\varepsilon(y)$ denote the open ball around $y$ with radius $\varepsilon>0$ $\omega_n$ denote the volume of the $n$-dimensional unit ...
3
votes
1answer
42 views

Prove $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$

Show that for bounded $X$ and $Y$ that $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$. Attempt: Suppose that $X = _{\mathcal{X}_F}$, where $F \in \mathcal{D}$. Then for every $B \in ...
1
vote
2answers
43 views

How to find $\sigma$-algebra for given that a function $f:X\to Y$ is measurable?

Let $X=\{1,a,b,c\}$, $Y=\{2,A,BC\}$ be two sets, and $f:X\to Y$ be a function such that: $f(1)=2$, $f(a)=A$, and $f(b)=f(c)=BC$. Choosing $\mathcal{G}=\{\underbrace{\{2\}}_{\textrm{set~}G_1}, ...
1
vote
1answer
8 views

Finding $\sigma$-algebra for given its generator.

Let $X=\{1,a,b,c\}$, $Y=\{2,A,BC\}$ be 2 sets, and let $f:X\to Y$ be a function such that: $f(1)=2$, $f(a)=A$, and $f(b)=f(c)=BC$. We have a collection ...
3
votes
2answers
37 views

Why sigma algebra and not other systems of sets?

I got a short question. Why we are considering sigma-algebras and not other systems of sets in measure-theory? Why not dynkin-system for example?
6
votes
1answer
49 views

Filtrations and Sigma-Algebras and Stopping Times

In a previous post Filtrations and Sigma-Algebras I asked the question: $\textbf{Previous Question:}$ Let $\Omega=\{1,2,3\}, \mathcal{A}=\mathcal{P}(\Omega)$ and $P(\{\omega\})=\tfrac{1}{3}$ for each ...
3
votes
1answer
28 views

The Lebesgue outer measure

The Lebesgue outer measure on $\mathbb{R}$ is defined as: $\lambda^{*}(A)$ = $inf${$\sum_{n=1}^{\infty}(b_{n}-a_{n}): A \subset \bigcup_{n=1}^{\infty}(a_{n}, b_{n}) $} I want to show that ...
1
vote
1answer
26 views

Showing 1/g is a measurable function

Let $g:(X, \mathcal{A}) \rightarrow \overline{\mathbb{R}}$ be a measurable function. Then could anyone give me a simple proof on how to show $1/g$, $g \neq 0$ is a measurable function?
1
vote
0answers
18 views

Convergence in measure problem

Let $f_n=\sqrt{x^4+\frac{x}{n}}$ and $f(x)=x^2$ be functions on $(0,\infty)$. Prove that $f_n \to f$ in measure on $(0,\infty)$ and $f_n^2 \nrightarrow f^2$ in measure on $(0,\infty)$. Here the ...
4
votes
1answer
25 views

Show that $\sup f_n = \sup g_n$ almost everywhere.

Suppose that we have a sequence of (Lebesuge) measurable functions $f_n,g_n$ from $\mathbb{R}$ to $\mathbb{R}$. Suppose also that for each $n$, $f_n=g_n$ almost everywhere. I want to prove that ...
0
votes
0answers
29 views

Extension of the duality of the space of distributions over $X$ locally profinite space

I have two questions on a definition that appears in the book "Répresentations des groupes réductifs $p$-adiques" by David Renard (http://www.math.polytechnique.fr/~renard/Padic.pdf). Let $X$ be a ...
1
vote
1answer
37 views

Wrong proof: The outer measure of [0,1]= 0

The definition of outer measure of a set $E$ in $\mathbf{R}$ is: $m_*(E)=inf\Sigma|I_j|$ where ${I_j}$ the family of open intervals that cover $E$. I am wondering why this proof is wrong: $\forall ...