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

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Atomless measure space without measure preserving isomorphisms

Question: Could somebody give an example of an atomless measure space without measure preserving isomorphisms (except for the identity)? Background: A measure preserving isomorphism on a measure ...
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1answer
28 views

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

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
11 views

Subnormal Weighted shift and First order derivative

Let $\mathbb B^m$ denote the Eucledian ball in $\mathbb C^m.$ Does there exist a reinhardt measure $\mu$ supported on $\partial \mathbb B^m,$ the boundary of ball, so that the Hilber space $H^2(\mu)$, ...
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1answer
28 views

Interchanging of order summation in proposition 1.25 [Rudin RCA]

Hello! This proposition from Rudin's RCA book. One moment confuses me, namely how he interchanges the order of summation in that double infinite series? Can anyone give a rigorous explanation of it? ...
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1answer
28 views

Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
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0answers
26 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
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0answers
24 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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2answers
37 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
29 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 ...
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2answers
29 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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0answers
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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
22 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
31 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 ...
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1answer
41 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 ...
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0answers
19 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 ...
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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 $\forall \epsilon>0, \lim_{n\to\infty} \mu (\{x\in X\big | |f_n (x)-f(x)|\geq \epsilon\})=0$. ...
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1answer
33 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 ...
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2answers
52 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 ...
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1answer
27 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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3answers
84 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 ...
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1answer
36 views

Lebesgue integral, path connected and compact

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 ...
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1answer
18 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 $\mu(G|H)<\epsilon$.
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1answer
38 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, ...
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0answers
27 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 ...
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1answer
17 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, ...
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1answer
46 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? By non-trivial I mean that no set exists of cardinality equal to that of the ...
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Convergence of $\chi_{A_n}$ to $\chi_A$, where $A$ is the union of the sets $A_n$ [closed]

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 ...
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0answers
23 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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2answers
81 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 ...
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0answers
24 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 ...
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0answers
29 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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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 ...
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0answers
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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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1answer
24 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) $.
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Why is this sequence of functions uniformly integrable and tight? [closed]

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?
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0answers
29 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 ...
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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 ...
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21 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 ...
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0answers
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$\{ \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.? [closed]

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 ...
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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 ...
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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 ...
4
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2answers
148 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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A question regarding Vitali Covering Covering Lemma

Let f be an increasing function on the closed, bounded interval [a, b]. Define $$E_{\alpha}=\{ x\in(a,b) | \bar{D}f(x) \geq \ \alpha \}$$ Choose $\alpha^{'} \in (0, \alpha)$. Let $F$ be the collection ...
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1answer
22 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} ...
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2answers
32 views

Measurability of the image of a continuous function

I am working on this exercise. Assume that $f$ is continuous on $[a.b]$. Let $m$ be Lebesgue measure, and assume that $f: [a,b] \rightarrow \mathbb{R}$. Prove that $f$ satisfies the condition ...
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1answer
12 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
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1answer
19 views

Explain the use of Dominated Convergence Theorem

In the proposition below from Measure Theory and Probability by Athreya and Lahiri, DCT was used to justify the existence of $t$ in the first line of the proof. But I can't think how this was ...
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1answer
88 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
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0answers
29 views

Two Similar Measures on a Probability Space

Let $(\Omega,\mathscr{F}, P)$ be a probability space and let $Q$ be another probability measure on $\mathscr{F}$, and let $\mathscr{F}_n=\sigma(Y_1,\ldots,Y_n)$ be a non-decreasing sequence ...
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0answers
30 views
+50

algebraic sum of a graph of continuous function and itself - measure > 0 imply nonempty interior?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Then $G+G$ is compact: algebraic sum of a graph of continuous function and itself Borel or ...