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

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4
votes
1answer
29 views

Is $\nu(E) = \max\{\mu(E),\eta(E)\}$ a measure

Let $\mu$ and $\eta$ be measures on the measurable space $(X,\mathcal{M})$. For $E \in \mathcal{M}$, define $\nu(E) = \max\{\mu(E),\eta(E)\}$. Is $\nu$ a measure on $(X, \mathcal{M})$? My Try: I use ...
1
vote
2answers
38 views

Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
0
votes
1answer
24 views

Proving the impossibility of a particular binary sequence

Let $\Omega = \{0,1\}^{\mathbb{N}}$. My question is as follows. Can there exist an $\omega \in \Omega$ such that $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \omega_{k+a} = \frac{1}{2} \qquad ...
0
votes
0answers
35 views

Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow f$ a.e. [duplicate]

Let $m$ be Lebesgue measure on $\mathbb R$ and let $f_n ,f \in L^1 (m)$. Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow ...
0
votes
0answers
15 views

a sum is convergent to a function measurable Lebesgue

Show that the series $$\sum_{n\ge 1} \frac{\sin(nx)}{n}$$ is convergent on the set of real numbers to a function which is measurable Lebesgue. I dont really know how to deal with that..can you give me ...
0
votes
0answers
28 views

Banach Tarski proof understanding

Theorem (Banach-Tarski Paradox): The unit ball $\mathbb{D}^3 \subset \mathbb{R}^3$ is equi-decomposable to the union of two unit balls. Proof: Let $\mathbb D^3$ be centered at the origin, and $D^3$ ...
0
votes
1answer
26 views

$L^{p}$ identity for the maximal operator.

This is probably a very easy question, but I'm failing to understand it. Given a function $f \in L^{p}(\mathbb{R}^n)$, $1<p\leq \infty$, we define the uncentered maximal function of $f$ as $$ ...
2
votes
0answers
41 views

Linear operator on $L^p$

Let $\{E_1, E_2, ..., E_n\}$ be a collection of pairwise disjoint measurable subset of $(\mathbb{R}, m)$ ($m =$ Lebesgue measure on real line) with $0 < m(E_i) < \infty.$ Then for each $1 \leq p ...
1
vote
0answers
21 views

Measure of a set determined by a measurable function

Let $f$ be a function $f:[0,1]\to\mathbb{R}$, $f(0)=1$ and $f(x)= \sqrt{x}\cos(\frac{1}{x}) $, if $x \neq 0$. I have to find out the Lebesgue measure of the set: $$A=\{x\in[0,1]~~\mid~~ f(x)<0\}$$ ...
2
votes
0answers
33 views

Property derived form Monotone Convergence Theorem

Assume $\mathbb{E}|X_1| < \infty$ and $X_n \uparrow X$ a.s., then Monotone Convergence Theorem either provide $\mathbb{E}X_n \uparrow \mathbb{E} X < \infty$ or else $\mathbb{E}X \uparrow \infty$ ...
1
vote
0answers
32 views

Fubini's Theorem and expectation of random variables

I have a question regarding the application of the Fubini's Theorem to the expectation of the product of two random variables. Let $X,Y$ be two random variables defined on the probability space ...
3
votes
1answer
24 views

$\sigma$-field generated by the continuity sets of a measure

Let $\mu$ be a probability measure on the Borel subsets of a topological space $X$ (a compact metric space if necessary). A Borel set $B$ is a $\mu$-continuity set if $\mu(\partial B)=0$, where ...
1
vote
2answers
68 views

Additivity of the union of Lebesgue measurable sets [closed]

Please help me with this!! I dont have any idea how to use the information given in the hypothesis: If $(A_{n})_{n\ge 1}$ is a family of measurable Lebesgue sets, with $\lambda(A_{n}\cap A_{m})=0$, ...
2
votes
0answers
41 views

Interior of a Minkowski sum satisfies Brunn-Minkowski inequality.

If $A,B$ are Lebesgue measurable sets in $\mathbb{R}^n$ with $0 <\lambda(A),\lambda(B)< \infty$. Prove that $\lambda (\text{Int}(A+B))^{1/n} \ge \lambda(A)^{1/n}+ \lambda(B)^{1/n}$ where ...
1
vote
0answers
13 views

An exercise on image measure

Given: $((0,1), \mathbb B((0,1)), \lambda)$: a measure space, where $\lambda$ is Lebesgue measure. $\mu$: a probability measure on $(\mathbb R, \mathbb B(\mathbb R))$ $F(t) = \mu((-\infty,t])$, ...
1
vote
0answers
40 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
1
vote
1answer
20 views

A one-sided continuous stochastic process is product measurable. Does the same hold true for almost surely continuous processes?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(X_t)_{t\ge 0}$ be a real-valued almost surely continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$. Let ...
1
vote
1answer
41 views

$\left\|f\right\|_{L^1(μ_1)}<∞$ $μ_2$-a.e.,$\left\|f\right\|_{L^1(μ_2)}<∞$ $μ_1$-a.e. $⇒$ $\left\|f\right\|_{L^1(μ_1\otimesμ_2)}<∞$

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space and $f:\Omega_1\times\Omega_2\to\mathbb R$ be measurable with respect to $\mathcal A_1\otimes\mathcal A_2$. Can we conclude, that ...
2
votes
0answers
28 views

Any rule of thumb that says any reasonable function I can write down is measurable?

Question arose in the context of probability: If $Y$ is $F-$measurable and $h$ is some function, $h(Y)$ is $F-$measurable if $h$ is a measurable function (Doob-Dynkin). For example, let $f(x,y)$ be ...
2
votes
1answer
23 views

Fubini-Tonelli theorem and absolutely Lebesuge integrable functions

As far as I know, a measurable function is Lebesgue integrable if and only it is absolutely integrable. It is simply because the definition of the integrability requires each of the positive part and ...
5
votes
0answers
54 views

Can a Banach measure be consistent with Friedman's Fubini-type theorem for non-measurable functions?

I read on Wikipedia that Harvey Friedman proved that the following is consistent with ZFC+¬CH: For all functions $f:[0,1]^2 \mapsto \mathbb{R}^+$ such that both $\int_0^1 \left ( \int_0^1 f(x,y) dy ...
1
vote
0answers
25 views

looking for help with convergence in measure problem

Why is it true that if $(X, \mu)$ is a finite space, $f_n \to f$ in measure, and for each $n$ and $\epsilon > 0$ there exists $\delta > 0$ such that $\mu(E) < \delta \Longrightarrow \int_E ...
2
votes
0answers
36 views

Union of uncountable family of $\sigma$-algebras

Suppose that $\{F_\alpha\}$ is an uncountable family of $\sigma$-algebras, and let $H=\bigcap_\alpha F_\alpha$. Is $H$ also a $\sigma$-algebra? Why or why not? I understand that the intersection ...
0
votes
0answers
15 views

Ordering on probability measures implies equality

Let $\Omega$ be a non-empty set, $\mathcal{F}$ a $\sigma$-algebra of subsets of $\Omega$ and $P,~Q$ two probability measures on $(\Omega, \mathcal{F})$. Assume that $P(A)\leq Q(A)$ for all $A\in ...
2
votes
0answers
62 views

Will the generated sigma algebra have this property?

Lets say you have a measurable space $(\Omega, \mathcal{A})$. And a measurable function $X: (\Omega, \mathcal{A})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We then know that for the sigma ...
3
votes
2answers
46 views

For a.e. $x \in [0, 1]$, there are finitely many $p/q$ such that $\left| x - p/q \right| < 1 / \left( q \log q \right)^2$

I am stuck on a qualifying exam problem and was hoping to get some help. Show for a.e. $x \in [0, 1]$ that there are finitely many $p/q \in \mathbf{Q}$ in reduced form such that $q \geq 2$ and ...
0
votes
0answers
19 views

When $X_{n\wedge N}$ converges to $X_N$ in probability for martingale $X_n$ and stopping time $N$?

Suppose $\sigma$-algebras $\{\mathcal{F}_n\}$ is a filtration and random variables $\{X_n\}$ are adapted to $\{\mathcal{F}_n\}$. $N$ is a stopping time w.r.t $\{\mathcal{F}_n\}$. If $(X_n, ...
2
votes
1answer
37 views

Exercise about measurable and continuous functions

I want to propose to you this exercise. Let $f:[0,1]\times \mathbb{R}\to\mathbb{R}$ a function with these properties: 1)For every $x\in\mathbb{R}$ the map $t\mapsto f(t,x)$ is measurable. 2)For ...
1
vote
0answers
22 views

Convergence in measure : some questions.

I know that $f_n\to f$ in measure if $$\forall \varepsilon>0, \lim_{n\to\infty }m\{x\mid |f_n(x)-f(x)|>\varepsilon\}=0.$$ Does it mean that: Q1) $f_n\to f$ in measure if $$m\left\{x\mid ...
1
vote
1answer
35 views

If $f_n\to f$ in measure, is there a subsequence s.t. $f_{n_k}\to f$ a.e.? [duplicate]

If $f_n\to f$ in measure, is there a subsequence s.t. $f_{n_k}\to f$ a.e. ? The convergence in measure is a little be abstract to me, I don't really see what it means (even if I know the definition). ...
0
votes
0answers
16 views

Convergence of a sequence of measure distribution functions

I'm trying to show that for $f$ and $f_n$ measurable, $|f_n|\uparrow|f|$ implies $d_{f_n}\uparrow d_f$, where $d_f(\alpha)$ is the distribution function given by $$d_f(\alpha):=\mu(\{x: ...
0
votes
1answer
27 views

Integration respect to two measures

Let $(X,\mathcal{K},\mu_1)$ and $(X,\mathcal{K},\mu_2)$ be two measure spaces, and let $\mu=\mu_1+\mu_2$. Assume $f:X\to[0,+\infty]$ are integrable both respect to $\mu_1$ and $\mu_2$, how do we ...
0
votes
0answers
25 views

Is every strongly measurable function to $\mathbb{R}$ also measurable?

Here seems a simple enough proof of the statement: Let $f: X \to \mathbb R_{≥0}$ be measurable. Construct $$f_n(x)=\sum_{k=0}^{n^2}\frac{k}{n}\cdot ...
3
votes
0answers
26 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
0
votes
3answers
18 views

Outer Measure definition

In the definition of Lebesgue outer measure/ outer measure , $m^*(A) = inf \{\sum l(I_n)\}$ Here how can one take infimum over a summation? Please elaborate.
2
votes
1answer
28 views

Random function of random variable

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Suppose that $\phi:(\Omega,\mathbb{R})\rightarrow\mathbb{R}$ is such that $\phi(\omega,\cdot)$ and $\phi(\cdot,x)$ are Borel measurable. ...
1
vote
1answer
25 views

Almost everywhere equality of r.v.'s , based on information on mean values.

Let $X, Y$ be two random variables on a probability space $(\Omega, \mathcal{F},P)$, where $\mathcal{F}=\sigma(\mathcal{E})$. We assume that: $\mathcal{E}$ is closed on intersections, i.e. $A\cap ...
1
vote
1answer
31 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
0
votes
0answers
31 views

Real Analysis, Folland problem 3.2.14 The Lebesgue - Radon-Nikodym Theorem

Relevant background information: We say that two signed measures $\mu$ and $\nu$ on $(X,M)$ are mutually singular if there exists $E,F\in M$ such that $E\cap F = \emptyset$, $E\cup F = X$, $E$ is ...
0
votes
0answers
47 views

Understanding the concept of measurability of random variables

If a random variable $X$ is $\mathcal{F}_{t_0}$-measurable, where $\{ \mathcal{F} \} _{ t \geq 0}$ is an underlying filtration, does that mean that from the time $t_0$ onwards, the random variable $X$ ...
1
vote
0answers
35 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in ...
0
votes
1answer
42 views

Continuous Function on a Set With Content Zero

I am trying to prove a proposition about a continuous function over part of a compact set, and I have gotten stuck. The proof will be completed if I can verify the following: If $f$ is a continuously ...
7
votes
1answer
55 views

Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $

I have never seen an integral like this: $$\int (y-f(x))^2 Pr(dx,dy) $$ What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical ...
0
votes
0answers
17 views

“Equidecomposable”: informal meaning

I am having trouble understanding the definition of the term "equidecomposable". Is it like two sets are split into many sets and then these many sets can be joined together to make either of the two ...
2
votes
2answers
41 views

Lebesgue Integral of a non negative piecewise function

Consider the function over [0,1] given by $f(x)= \begin{cases} 0 & x \in \mathbb{Q}\\ x & x \notin \mathbb{Q} \end{cases}$ In order to compute the Lebesgue integral of $f$ we need to find an ...
2
votes
0answers
56 views

How to understand $E(X\mid B)$ in the measure theory way

From undergraduate probability course, we learn $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ given $P(B)>0$. And we learn that if $(X,Y)$ has a joint density $f(x,y)$, we can calculate marginal density ...
0
votes
1answer
15 views

Real Analysis, Folland Problem 3.3.20 Complex Measures

Related definitions - A complex measure on a measurable space $(X,M)$ is a map $\nu: M\rightarrow\mathbb{C}$ such that i.) $\nu(\emptyset) = 0;$ ii.) if $\{E_j\}$ is a sequence of disjoint sets in ...
1
vote
1answer
41 views

Book Recommendation for Measure Theory in n-Space

What's a standard book on multidimensional measure theory? I'm aware of some books on functions of several variables, but they do not discuss measure theory or Lebesgue integration in space. Thanks. ...
4
votes
1answer
58 views

Integration over finite partition of integration domain

I think the title does not reflect my problem very well. Feel free to leave a comment with a more appropriate title. Let $f \in L^1([0,1])$. How do I prove there exists a partition of $[0,1]$ into ...
1
vote
2answers
35 views

Why a cover of a set exist?

In the definition of the measure, we have that $$m^*(E)=\inf\left\{\sum_{i=1}^\infty |Q_i|\mid E\subset\bigcup_{i=1}^\infty Q_i\right\}$$ where $Q_j$ are closed cube. My question is : Why for any $E$ ...