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

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2
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1answer
33 views

Show that $\mathfrak{S}=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\}$ is a semi-ring

Let $\Gamma$ be a finite set, $\Omega=\Gamma^{\mathbb{N}}=\left\{(x_1,x_2,\ldots):~\forall i\in\mathbb{N} x_i\in\Gamma\right\}$. For $a_1,\ldots,a_N\in\Gamma$ let $$ ...
1
vote
0answers
195 views

Question on stochastic process

let $(\Omega, \mathcal{F},\pi)$ be a probability space with $\sigma$-algebra $\mathcal{F}$ and measure $\pi$. Let $$X:[0,+\infty)\times \Omega\rightarrow \mathbb{R}$$ a family of random variables ...
1
vote
1answer
99 views

When does a measurable function exist with a given distribution?

Let's suppose (A,X,P) and (B,Y,Q) are two probability spaces (A,B underlying spaces, X,Y sigma-algebras, P,Q probability measures, respectively). Under what (topological and/or measure theoretic) ...
0
votes
1answer
118 views

Show that set $\{x\in]0,1[:f(x)=\alpha\}$, $\forall \alpha\in\mathbb{R}$ is measurable but $f$ is not.

This is a exercise of my course in Measure and Integration. I did one part but the end is confounding me and I don't know do. The exercise We view in class that if $E$ is a measurable set in ...
0
votes
1answer
70 views

If $F\subset G$ where $F$ is a closed set and $G$ is a open set, show that $m^*(G\backslash F)=m^*(G)-m^*(F)$.

The exercise is: If $F\subset G$ where $F$ is a closed set and $G$ is a open limited set, show that $m^*(G\backslash F)=m^*(G)-m^*(F)$. I try solve this: If $G=F\dot{\cup}(G\backslash F)$, ...
5
votes
3answers
150 views

If $f_{n}$ are non-negative and $\int_{X}f_{n}d\mu=1$ does $\frac{1}{n}f_{n}$ converge almost-everywhere to $0$? does $\frac{1}{n^{2}}f_{n}$?

Question from an exam sample I'm studying for: Suppose $\left(X,\mathcal{F},\mu\right)$ is a measure space and $f_{n}:\left(X,\mathcal{F}\right)\to\mathbb{R}$ a sequence of non-negative integrable ...
1
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0answers
43 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
4
votes
1answer
487 views

Show that if $E\subset\mathbb{R}$ is a measurable set, so $f:E\rightarrow \mathbb{R}$ is a measurable function.

If $E\subset \mathbb{R}$ is a set Lebesgue Measurable and $f:E \rightarrow \mathbb{R}$ a monotone function, show that $f$ is measurable. I'm trying for hours with no progress.
1
vote
1answer
77 views

Showing that the hypothesis that $m (E) <\infty $ is essential in the Egoroff's theorem.

In Egoroff theorem, the hypothesis that $ m (E) <\infty $ is essential. Construct an example of measurable functions $ f_n: \mathbb{R} \rightarrow \mathbb {R} $ that converge to the null function ...
1
vote
1answer
42 views

Is $\int g\, d\mathbb{P}=\int g\, d\mathbb{P}_{|\mathfrak{F}}$?

Let $(\Omega,\mathfrak{A},\mathbb{P})$ be a probability space and $\mathfrak{F}\subset\mathfrak{A}$ a sub-$\sigma$-algebra. Consider $g\in L_{\mathbb{P}}^1$ and $g$ $\mathcal{F}$-measurable. ...
2
votes
2answers
89 views

$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) ?$

In my functional analysis script there is an example that uses $$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) $$ where $\Omega \subset \mathbb{R}^n$ is an open subset and we take $L^{\infty}$ ...
0
votes
1answer
61 views

Edited: Defining a measurable pointwise limit for a sequence of measurable functions.

Edit: I edited the post considerably to focus attention to the primary question Let $\left(X,\mathcal{F},\mu\right)$ be a measure space and let $f_{n}:\left(X,\mathcal{F}\right)\to\mathbb{R}$ be ...
1
vote
1answer
100 views

$L\log L$ and $L^p$ embedding

My question is a simple one: I am aware of the embedding $L^p(\Omega)\in L\log L(\Omega)$ for finite measure spaces, with constant $\frac{cp}{p-1}$. Does this embedding hold on for instance, the whole ...
10
votes
1answer
609 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
1
vote
1answer
163 views

Lp space over measure space and over its completion is the same.

It is exercise of Tao's 1.3.1 Let $(X, \chi,\mu)$ and $(X, \overline{\chi}, \mu)$ is a measure space, where $\overline{\chi}$ is the completion of $\chi$. Show that $L^{p}(X, \chi, \mu)$ is ...
1
vote
2answers
206 views

Understanding the definition of the integral of a nonnegative function in measure theory

Let me begin by providing the following two definitions from my class notes. I was trying to put together how from the definition of a simple function, we would form the given definition of an ...
1
vote
1answer
55 views

understanding pointwise convergence a.e. in measure theory

I'm going through a proof and the assumptions are that $\mu$ is a complete measure and that $f_n\rightarrow f$ $\mu$-a.e. One of the lines in the proof says If $f_n\rightarrow f$ $\mu$-a.e., then ...
1
vote
1answer
55 views

Show: The integral over a zero set is zero

From Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$ was motivated: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space. Let $A\in\mathfrak{A}$ with $\mu(A)=0$. I would like to show that $\int_A f\, ...
2
votes
1answer
190 views

Question related to Egorov theorem

Let $(X,\mathcal{F},\mu)$ be a finite measure space, and $(f_k)_{k=1}^{\infty}$ be a sequence of measurable real valued functions with $$\lim_{k\to\infty}f_k(x)=0\quad\mu-a.e.$$ Use Egorov theorem ...
4
votes
1answer
64 views

Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathcal{F}$ a sub-$\sigma$-algebra. Let $\mathcal{F}$ be trivial, i.e. $\forall A\in\mathcal{F}: ...
2
votes
1answer
100 views

Is $fg\in L^1$ when $f,g\in L^1$?

Let $(X,\mu)$ be a measure space. Let $f,g:X\rightarrow \mathbb{F}$ be functions in $L^1(\mu)$. ($\mathbb{F}$ is the field of real or complex numbers). When I learned Riemann-Integral, it was quite ...
1
vote
1answer
208 views

A basic property of the Lebesgue outer measure

If $G$ is a measurable set and satisfies $m^*(G)<\infty$, then for all $\varepsilon>0$ there exists a closed set $F\subset G$ such that $m^*(F)>m^*(G)-\varepsilon$ Edit: I know that: ...
0
votes
2answers
361 views

A function which is continuous in one variable and measurable in other is jointly measurable [closed]

Please help me to prove the following; Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ f(x,\cdot)$ is measurable for each fixed $x\in[0,1]$; (ii) $\ f(\cdot,y)$ is ...
1
vote
2answers
50 views

$\{ (x,x) ∣ x \in B\}$ is measurable if and only if $B$ is measurable

How to prove the following If $B \subseteq \mathbb{R}$ then $\{ (x,x) ∣ x \in B\}$ is Borel measurable if and only if $B$ is Borel measurable Specifically how do you prove the implication, if ...
1
vote
1answer
66 views

Measurable Functions $g(x)=f(x^2)$

$f: {\mathbb R} \rightarrow{\mathbb R}$ If $g: {\mathbb R} \rightarrow{\mathbb R}$, $g(x)=f(x^2)$ measurable, then $f$ is also measurable? I try to use the definition to measurable functions with ...
1
vote
1answer
34 views

If $A$ is measurable then so is $\alpha A$

I'm struggling to show this fact. I have already shown that $m(\alpha A) = |\alpha|A$ and I now need to show that $\alpha A$ is measurable. So far I have: Take an arbitrary subset $E \subset R$ then: ...
5
votes
0answers
88 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
0
votes
1answer
43 views

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Let $-\infty<a<b<\infty$ and $f,g\in H^1(a,b)$. So, $f,f',g,g'\in L^2(a,b)$. Suppose $$\int_a^b|f+g'|^2\mathrm dx=\int_a^b|f'-g|^2\mathrm dx=0.$$ Is it possible to conclude that $f=g=0$ ...
6
votes
1answer
473 views

Example when triangle inequality fails in weak $L^p$ spaces

Let's consider the quasi-norm on the weak $L^p(X,m)$ spaces: $$[f]_p=\sup_{t>0}\left\{t\, m\big(\{x:|f(x)|>t\}\big)^{1/p}\right\}.$$ We know that it is not a norm since the triangle inequality ...
5
votes
2answers
4k views

Example of $\sigma$-algebra

I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the ...
-1
votes
1answer
75 views

Equivalence of these three statements about convergence of events in a $\sigma$-algebra

Could you explain to me why these three statements are equivalent. Everything takes place in a $\sigma$-algebra $\mathcal A$. Let $\{A_n\}$ be a sequence of events in $\mathcal A$. $1_A$ denotes the ...
0
votes
1answer
199 views

Using hint to prove a result about the Lebesgue outer measure.

I'm trying to solve this but without success. The Question Prove that: if $F_1$ $F_2$ are bounded closed sets in $\mathbb{R}$, so $m^*(F_1\cup F_2)=m^*(F_1)+m^*(F_2)$ where $m^*$ denote the ...
0
votes
1answer
54 views

$\sigma$-field and family of sets

Let $f:[0,1] \to [0,1]$ be the function $f(x)=-4x^{2} + 4x$ let $\mathcal M$ be any $\sigma$-field in $[0,1]$. Is the family of sets $\mathcal N= \{f(A): A \in \mathcal M \} $ a $\sigma$-field in ...
3
votes
1answer
52 views

algebra generated by a countable set is countable

How to prove that algebra generated by a countable set is countable. Hint enough. I know that algebra generated by any set $A$ is of the form $\cup_{i=1}^{m}\cap_{j=1}^{n_i} A_{ij}$ where $A_{ij}$ or ...
0
votes
2answers
102 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
0
votes
1answer
55 views

need help with proving unboundedness of a linear functional

Let $X$ be a normed linear space and let $F$ be a linear functional defined on $X$. Prove that $F$ is unbounded if and only if for each $a \in X$ and each $r > 0$, $\{F(x) : ||x- a|| < r\} = R$. ...
0
votes
1answer
60 views

Limit of decreasing functions

Let $f_n$ be a sequence of positive, integrable, decreasing functions. Assume Lebesgue measure. Suppose $\int f_n$ goes to $0$. Show $f_n$ goes to $0$ almost everywhere. This is a Qualifying exam ...
3
votes
1answer
982 views

Inverse image of $\sigma$-algebra

Is this proof correct? Let $f$ be a function mapping $\Omega$ to $E$ with $\mathcal E$ a $\sigma$-algebra on $E$. Show that $\mathcal A=\{f^{-1}(B):B\in \mathcal E\}$ is a $\sigma$-algebra on ...
3
votes
1answer
274 views

Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
2
votes
2answers
91 views

Sub $\sigma$-algebra

Is my proof of the following correct? Let $\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ and let $B\in\mathcal{A}$; then $\mathcal{B}=\{A\cap B:A\in\mathcal{A}\}$ is a $\sigma$-algebra on $B$ ...
1
vote
1answer
158 views

Show that sequence converges pointwise to a function that is not Riemann Integrable.

This is an exercise of the course of Measure and Integration and I'm having trouble to solve this. I not know how to show the sequence is of Cauchy and why are not R-Integrable. Let the sequence ...
2
votes
2answers
62 views

Rate of appearance of digits 7 and 8 in a given sequence

This is from Arnold's Mathematical Methods of Classical Mechanics: let us consider the sequence $$ 1,\ 2,\ 4,\ 8,\ 1,\ 3,\ 6,\ 1,\ 2,\ 5,\ 1,\ 2\ \ldots $$ which consists of the first digits of the ...
1
vote
1answer
70 views

Adaptedness of random variables

Suppose we have an RCLL adapted process $(X_t)$. Moreover we are given a stopping time $T$. We define $\mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}$. ...
3
votes
1answer
91 views

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

in preparation for an exam I wanted to show that for $\mu(\Omega)=1$ and $h:\Omega \rightarrow [0,\infty]$ measurable the following inequality holds: $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq ...
1
vote
2answers
50 views

understanding simple functions

Let $(X,\mathcal{M})$ be a measurable space. The definition of a simple function on a set $X$ is that it is a finite linear combination, with real coefficients, of characteristic functions of sets in ...
4
votes
1answer
227 views

A question about Jordan measure.

(a) Suppose that $A \subset [a,b]$ and that exists a partition $P$ of $[a,b]$ such that $c_e(A;P)<\eta$. Show that exists $\delta>0$ such that, if $Q$ is a partition where $|Q|<\delta$ so ...
5
votes
2answers
98 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...
1
vote
1answer
58 views

Equivalence of measurable functions

I want to prove the following Lemma: Let $\mathcal{A}$ be a $\sigma$-algebra in $X$ and let $f:X\rightarrow\mathbb{R}$, then TFAE: $f$ is measurable. For each Borel set $B\subset\mathbb{R}$ holds ...
2
votes
0answers
134 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
2
votes
1answer
7k views

Bonferroni inequality proof

Is this proof for $P(\bigcup_{i=1}^n A_i)\le\sum_{i=1}^nP(A_i)$ correct? Pf. By induction. For $n=2$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)\le P(A)+P(B)$$ Assume that the statement is true for $n-1$, ...