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

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36 views

When a limsup can enter inside an integral

In the book "partial differential equation in classical mathematical physics" by I.Rubinstein,L.Rubinstein at page 411 I found something that I can't justify. It seems that the author (between ...
1
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0answers
29 views

Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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4answers
35 views

Relation between counting measure and Tonelli theorem

This is from Rudin's RCA book. But I can't understand how he got Corollary. What he takes as $f_n, X$? If we consider counting measure how integral converts to sum? I can't show this after some ...
2
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1answer
40 views

Please check whether the proof is correct or not.

Please check my solving. I want to know where to be wrong or illogical, or where logical jumps are. Problem Let $y=Tx$ be a nonsingular linear transformation of $\mathbb{R}^n$. If ...
1
vote
1answer
32 views

$L_p(\mu)\subseteq L_q(\mu)$ [closed]

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?
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0answers
36 views

$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
2
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2answers
38 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
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0answers
8 views

measure concentrated on an axis

If I want to consider a Borel-measure $\mu$ in $\mathbb{R}^2$ concentrated on the x axis, how can it work? it could be $\mu(A) >0$, where $A \in \mathcal{B}(\mathbb{R}^2)$?
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0answers
58 views

Small filters are measurable

i want to show, that a filter $\mathcal{F}$ on $\omega$ (considered as a subset of $2^\omega$), which is small, is measurable. I found a lemma (without proof), that every small set is null. So, if ...
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0answers
5 views

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 ...
3
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1answer
36 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? ...
1
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1answer
29 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 $f \in C^1$ ...
2
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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
38 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
30 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 ...
2
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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
11 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
23 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
35 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 ...
2
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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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0answers
16 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
votes
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
53 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
29 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 ...
0
votes
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 ...
1
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1answer
21 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
28 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
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1answer
18 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, ...
4
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1answer
47 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 ...
-1
votes
0answers
34 views

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 ...
1
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0answers
24 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 ...
2
votes
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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0answers
26 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 ...
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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 ...
2
votes
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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0answers
19 views

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?
2
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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 ...
0
votes
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
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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votes
0answers
23 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.? [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 ...
0
votes
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 ...
1
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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 ...