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

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0
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25 views
+50

intersection of boundaries have Lebesgue measure 0

$A,B$ - sets in $R^n$ such that $\overline{A}\cap B \cup \overline{B}\cap A$ is empty then $bdry(A)\cap bdry(B)$ has $n$-dimensional Lebesgue measure 0. I don't know how to prove this fact. In ...
2
votes
2answers
94 views
+200

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
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0answers
11 views

Elements contain in a sigma algebra generated by a set of random variables

Hello and thanks for the time spend to read this :) Consider $(\Omega,\mathcal{F},P)$ Consider $A=\{x_1,...,x_p\}$ a set of random variables and $\Theta=\sigma(A)$ be the sigma algebra generated by ...
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0answers
16 views

Sequence of Radon Measures $\mu_n$ on $\mathbb{R}$

Problem: Find a sequence of signed Radon Measures $\mu_n$ on $\mathbb R$ such that $\langle \mu_n, \phi \rangle \to 0$ for every $\phi \in C^1_c(\mathbb R)$, and $|\mu_n|([0,1]) \to +\infty$. ...
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1answer
11 views

Random variable independent of $\sigma$-algebra and conditional expectation

What does it mean to say that a random variable is independent of a sigma-algebra, and why then does this imply that $E(RV| \sigma) = RV$?. I have no clue what this independence stuff is about ...
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1answer
17 views

Three questions on measurable functions and $L^p$ spaces

I'm learning about measure theory and $L^P$ spaces and need help with the following questions: True or False (justify): $(1)$ Let $f:(-1, 1) \to \mathbb{R}$ measurable on $(-n, n), \; \forall ...
7
votes
1answer
228 views
+100

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
0
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0answers
9 views

Every measurable subset of measure space is new measure space

Let $(X,\mathfrak{M},\mu)$ be a measure space and let $E\in \mathfrak{M}$. Prove that $E$ is also measure space. Proof: $(E,\mathfrak{M}_E,\bar\mu)$ be a measure space where $\bar \mu$ is "old" ...
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0answers
9 views

Conditional expectation and set times random variable??

On page 62, what in the world is the meaning of equation (5.2)? $\mathcal{F}_t$ is a $\sigma$-algebra, so $Z_t \in \mathcal{F}_t$ is a set. $X_u$ is a random variable, so what is $Z_t X_u$?
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1answer
14 views

$\mathcal{L}^N(B_r(x)\cap E)> 0 \hspace{0.6cm} \forall r>0$ if every point is a Lebesgue Point

Exercise: Let $E$ be a Borel set such that every point is a Lebesgue Point for $\chi_E$ , and let $x \in \partial E$ (the topological boundary). Show that $\mathcal{L}^N(B_r(x)\cap E)> 0$, and ...
0
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1answer
25 views

Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
0
votes
3answers
65 views

Prove that $\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$

Let $(X,\mathfrak{M},\mu)$ be measure space. Let $f\geq 0$ be measurable function. Prove the following equality: $$\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$$ I can show only that $\int ...
0
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1answer
32 views

Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
2
votes
0answers
14 views

$X_t$ measurable wrt $\sigma$-algebra and “revaled information”

Studying stochastic processes, it is mentioned that if $(X)_t$ is a process and $(\mathcal{X})_t$ a filtration, then if the process is adapted to the filtration, the informal way to think about it is ...
0
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0answers
15 views

Product of Lebesgue-null-set and arbitrary Lesbesgue-set is a Lebesgue-null-set again

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
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0answers
15 views

Interesting measure theory property in L^p [duplicate]

Let $f, f_n \in L^p (X)$, so that there is a function $g\in L^p (X)$ with $|f_n|\leq g,\ \forall n$ and $\forall \epsilon>0, \lim_{n\to\infty} \mu (\{x\in X\big | |f_n (x)-f(x)|\geq \epsilon\})=0$. ...
0
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0answers
23 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
0
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1answer
30 views

Lebesgue integral, path connected and compact function

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
1
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1answer
21 views

regularity of a measure

Let $\mathcal{A}$ be a $\sigma$-algebra containing the Borel algebra (everything is in a topological space). Let $m\colon\mathcal{A}\to[0,\infty]$ be a measure. The standard definition of regularity ...
1
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1answer
15 views

$A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$

Let A be a real set then is it true that $A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$.
2
votes
2answers
70 views

Real Analysis, Folland problem 1.4.24 Outer Measures

Let $\mu$ be a finite measure on $(X,M)$, and let $\mu^*$ be the outer measure induced by $\mu$. Suppose that $E\subset X$ satisfies $\mu^*(E) = \mu^*(X)$ (but not that $E\in M$). a.) If ...
0
votes
1answer
37 views

What is the value of the measure of a line segment?

Let $$f(x)=1-x^2$$ Then $$|\{x\in\mathbb{R^1}:f(x)>0\}|=|(-1, +1)| = 2$$ Let $f$ be a nonnegative function, defined on measurable subset $E$ of $\mathbb{R}^n$. Then $\Gamma(f, ...
0
votes
0answers
25 views

Linear functional and Riesz' Rep theorem

On page 59 in these Finance notes, a positive linear functional is defined, and then Riesz' representation theorem is used (the scalar product is defined on bottom part of page 56). I don't ...
0
votes
1answer
16 views

Atoms as partitions

Is every $\sigma$-algebra generated by a partition? In the answer, in the first paragraph, it is written that if a finite set is used to generate a $\sigma$-algebra, every point is in a unique atom, ...
3
votes
0answers
30 views

Nonatomic measure space over set larger than the reals

Question: Does anybody know a non-trivial nonatomic measure space over a set larger of cardinality larger than the reals? Background: I try to find a big list of examples of measure spaces, but I ...
5
votes
1answer
41 views

Let $f: [0, 1] \to \mathbb{R}$ s.t $f(0)=f(1)=0$ then measure of $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\} \geq 1/2$.

Let $f:[0,1]\to\mathbb R$ be a continuous function s.t. $f(0)=f(1)=0$. Let $$A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\}.$$ Show that set $A$ has Lebesgue measure $\geq 1/2$. ...
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0answers
34 views

Convergence of $\chi_{A_n}$ to $\chi_A$, where $A$ is the union of the sets $A_n$ [on hold]

Suppose $(X,\mathcal{M},\mu)$ is a measure space with $\mu$ is a complete measure. Let $f$ be a measurable function on measurable subset $A$ of $X$. Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of ...
1
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0answers
22 views

Monotone Convergence Theorem in Measure Theory.

My textbook defined M.C.T. by for $\{f_k\}$ be a sequence of measurable functions on $E\subset\mathbb{R}^n$, If $f_k\nearrow{}f~~a.e.$ on $E$ and there exists $\phi\in{}L(E)$ such that ...
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0answers
22 views

Finding the generating set of a $\sigma$ algebra

Let $\Omega=(0,1]$. Let $\beta$ be the Borel $\sigma-$algebra generated by open sets in $\Omega$. Now,$\tilde\beta$={$B\subset\Omega :B\in\beta$ and is either disjoint from$(\frac{1}{2},1]$ or ...
3
votes
1answer
112 views
+50

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
2
votes
1answer
23 views

Why is the discrete formulation of the fundamental theorem of integral calculus correct?

Define Diff$_hf = \frac{f(x+h)-f(x)}{h}$. Define Av$_hf(a)=\frac{1}{h}\int_a^{a+h} f$ Why is the following correct? $\int_a^b$Diff$_hf = Avf(b) -Avf(a) $.
0
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0answers
12 views

Given an event field, is there a random variable generating it? [duplicate]

In probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, for any event field $\mathcal{G}\subset\mathcal{F}$, there always exists a random variable $X$, such that $\sigma(X)=\mathcal{G}$? Is ...
1
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0answers
27 views

Definition of Lebesgue integral from Rudin RCA

Note that Rudin defines Lebesgue integral for function $f$ which is measurable. Is measurability is important here? What about if we'll define $(3)$ also for non-measurable function $f$?
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0answers
21 views

Exercise 8.O in Bartle's The Elements of Integration

I have a doubt about this exercise (8.O) in Bartle's book. Exercise 8.O I already answered the Exercise 8.N so I'm able to apply it, but, I just have no idea about how to do this. I'm working on ...
1
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0answers
17 views

Intuition behind Mutual independence of sub-$\sigma$-algebras definition.

I was reading about Independence of sub-$\sigma$-algebras when I found the next definition: Let $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ $n$ sub-$\sigma$-algebras of $\mathcal{A},$ let $H$ be a ...
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votes
0answers
19 views

Why is this sequence of functions uniformly integrable and tight? [on hold]

In a measure space (X, M, $\mu$) where M = {$\phi, E, X-E, X$} and $\mu(E)=\mu(X-E)=0.5\ \ \mu(X) = 1$ Define $f_n=n\chi_{E}-n\chi_{(X-E)}$. Why is $\{f_n\}$ uniformly integrable and tight?
2
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0answers
28 views

A detail on Fubini's theorem

Let $f(x, y)$ be a measurable function on a product of two balls $B_{1}$ and $B_{2}$ in $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ respectively and $m,n\geq1$. We know, according to Fubini's theorem, that ...
0
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1answer
18 views

Continuity of Integration (Lebesgue)

On the theorem regarding continuity of integration: Let $f$ be integrable over $E$. If $\{E_{n}\}^{\infty}_{n=1}$ is an ascending countable collection of measurable subsets on $E$, then ...
2
votes
1answer
813 views

Outer and inner approximation of set with finite outer measure

I was wondering if somebody could help me out with a solution for the following problem (taken from Royden's Real Analysis, 4e. (ch. 2.4, prob. 18): Let $E$ have finite outer measure. Show that ...
0
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1answer
13 views

If a simple function is nonnegative, why do the set on which the simple function is strictly positive have finite measure?

If a simple function is nonnegative, why do the set on which the simple function is strictly positive have finite measure? I know it should be Sigma finite, but why is it finite? This is from page ...
0
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0answers
18 views

Minimum Random Variables and Integration

We are given a sequence of independent random variables $\lbrace X_{nk} \rbrace$, for $k=1,...,r_{n}$, with $E(X_{nk})=0$ and $\sigma^{2}_{nk}<\infty$. My question involves a small piece of the ...
2
votes
1answer
41 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
0
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1answer
46 views

Why is a set function equivalent to its induced outer measure if it is countably monotone?

Denote Outer measure as $\mu^*(E)$ and a set function $\mu(E)$. Define outer measure as the following: μ*(E) = inf $\Sigma\mu(E_k)$ where $\{E_k\}$ cover E. Why is a set function equivalent to its ...
1
vote
1answer
85 views

Real Analysis, problem 1.4.22 Outer Measures

Exercise 22 - Let $(X,M,\mu)$ be a measure space, $\mu^*$ the outer measure induced by $\mu$ according to (1.12), $M^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, and $\overline{\mu} = ...
3
votes
2answers
144 views

Definition of positive measure

Why Rudin assumes that $\mu(A)<\infty$ for at least one $A\in \mathfrak{M}$? What about if $\mu(A)=\infty$ for any $A\in\mathfrak{M}$?
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votes
0answers
22 views

$\{ \int_E f_k(x)dx \}_{k\ge1} \to \int_E f(x) dx $ for any measurable set E, then $f_k(x) \to f(x)$ a.e.? [on hold]

Suppose $f, f_k \in L(R)$, and for any measurable set E, $\{ \int_E f_k(x)dx \}_{k\ge1} $ monotonically-increasingly converges to $\int_E f(x) dx$, Show that $f_k(x) \to f(x)$ a.e. ? Note that ...
3
votes
1answer
30 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the following result: If $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes ...
5
votes
1answer
70 views

Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal ...
1
vote
0answers
33 views

Details on Proving that $\lim_{n \rightarrow \infty}\int_{-M}^M f(x) \cos (nx) dx=0$ Using Density of Step Functions

I was working on a question very similar to this post: Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$ . I want to show that $\lim_{n \rightarrow \infty}\int_{-M}^M ...
1
vote
0answers
32 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...