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

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17 views

Measure theory theorem

So far I couldn't find theorems about equality of measures, I would appreciate book recommendations and help with this theorem. Let A be a family of subsets of Ω stable under intersection. If ...
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10 views

Integration with respect to variation

Can somebody please give me a reference to what concepts are used here (I think it might be lebesgue stieltjes integration but I can't find the instance of an integration of a function with respect to ...
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1answer
6 views

second Borel–Cantelli lemma (or converse result) application

I'm having issues with understanding how Borel-Cantelli lemma applies to the following exercise: If a coin is tossed infinite times, prove that the probability of getting 2 consecutive heads (or ...
2
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29 views

(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
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1answer
29 views

Integral relations when using different measures.

Let $(X,\mathcal{M})$ a measurable space and $\mu$,$\nu$ two non-negative measures s.t $\mu \geq \nu$. Does it hold that $\int_E f \, d\mu \geq \int_E f \,d\nu $ where $E \in \mathcal{M}$. I suspect ...
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1answer
31 views

If $\mu(A_n)\to 0$ then $\int_{A_n}f\to 0$

Let $(X,\Sigma,\mu)$ a measure space and $f\in L_p$, where $p\in [1,+\infty)$. Let $(A_n)$ be a sequence in $\Sigma$ such that $\mu(A_n)\to 0$. Then I want to prove that $\int_{A_n}fd\mu\to 0$. I ...
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0answers
26 views

Approximating simple function by continuous function

I am trying to solve this problem: If $\gamma$ is a simple function defined on $E \subset \mathbb R^d$, $E$ measurable, then there is $f:E \to \mathbb R$ continuous such that $$|\{x \in E: f(x) \neq ...
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1answer
30 views

How can I prove that $\int_X\left(\int_Y f_xdm_2\right)dm_1$ exists given the following conditions …?

Let $X=Y=[0,1)$ and $f(x,y)=\dfrac{1}{(1-xy)^a}$, where $a>0$, and $m_1=m_2$ the Lebesgue measure. I want to prove that $$\displaystyle\int_X\left(\int_Y f_xdm_2\right)dm_1$$ exists (the integral ...
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2answers
23 views

How can the Hausdorff measure be nonzero?

We have dim$F := \inf \left\{s > 0 : \mathcal{H}^s (F) = 0\right\}$. My question is, with dim$F$ defined as the value where the Hausdorff measure equals zero, then how can ...
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0answers
11 views

when is the maximum likelihood estimator measurable

For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood ...
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1answer
15 views

Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
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1answer
19 views

Is every $\sigma$-algebra also a semi-ring?

I wanted to know if every sigma algebra is a semi-ring? Looks like to me it is implied by the definition of a semi-ring. However I read many books and none of the books state that it is true.
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3answers
39 views

Counterexamples and convergences

I want to find counterexamples for the following "states": If $\int f_n\to \int f$ then $\int |f_n-f|\to 0$. If $\int |f_n-f|\to 0$ then $f_n\to f$ almost everywhere. Can you give me a hint of ...
1
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1answer
30 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
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0answers
25 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
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1answer
35 views

How to show that every set with Lebesgue outer measure zero is Lebesgue measurable?

Definition of Lebesgue measurable if for each $ε>0$, there exist a closed set $F$ and an open set $G$ with $F⊂E⊂G$ such that $m$ * $(G-F)<ε$. About this problem, $F$ can be a empty set that is ...
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3answers
24 views

Borel $\sigma$-algebra defintion question

So I am studying measure theory and I have found myself struggling to fully understand the concept of the Borel $\sigma$-algebra in depth. We know that the Borel $\sigma$-algebra is the smallest ...
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1answer
25 views

Difficulty with a differentiation of measures proof

This shows up in a proof about differentiating measures. I'm having trouble figuring it out: For any $x \in \mathbb{R}^n$, let $\mathcal{C}_r(x)$ denote the set of open cubes with diameter less than ...
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0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
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2answers
49 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
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0answers
14 views

Product of product-measurable function and measurable function product-measurable?

Given two measurable spaces $(\Omega, \mathcal{F}), (\Theta, \mathcal{F}_\Theta)$ and their product with the product-sigma-algebra $(\Omega \times \Theta, \mathcal{F} \otimes \mathcal{F}_\Theta)$ and ...
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4answers
81 views

What is the best book for measure theory? [on hold]

What is the best book about measure theory I want the book has a lot of solved exercise I don't want just definitions theorem and examples I want a book has exercise and solved exercise
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2answers
21 views

Does a.e. convergence imply the boundness in $L^1$?

Let $f_n : I = (0, 1) \to \mathbb{R}$ be a sequence of functions. If $$f_n \to 0 \;\; a.e$$ does it imply that $$f_n \;\; \text{is bounded in} \;\; L^1(I)?$$ Why yes/not? Thank you!
2
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1answer
41 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
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1answer
31 views

What does it mean that a sequence of functions is bounded in $L^1(I)$?

Let $I = (0, 1)$ and $f_n : I \to \mathbb{R}$ a sequence of functions. What does it mean that $f_n$ is bounded in $L^1(I)$? Does it mean that $$\exists c>0 \;\; \text{such that} \;\; \|f_n\|_1 ...
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0answers
12 views

Prove for measures $\mu $and $\nu$ $\nu \perp \mu$ iff $|\nu| \perp \mu$ iff $\nu^+ \perp \mu$ and $\nu^- \perp \mu$

Where $\perp$ means mutually singular. I have a question, as $\nu$ is clearly a signed measure do we assume that $\mu$ is signed or just positive? It follows from $\nu\perp\mu$ with the set in ...
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15 views

Algebra and $\sigma$-algebra

We consider 3 intervals $A_1$, $A_2$ and $A_3$, which are defined as $$ A_1=\left(-\infty,0\right], ~A_2=\left(0,\frac{1}{2}\right], ~A_3=\left(\frac{1}{2},+\infty\right). $$ We then form the ...
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0answers
24 views

A set of the second category has a positive measure?

A subset of a topological space $X$ is called nowhere dense in $X$ if the interior of its closure is empty. A subset of a topological space $X$ is called the first category (or meagre) in $X$ if it ...
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1answer
23 views

Doubt on Caratheodory's extension theorem

This doubt is on the Caratheodory's extension in Billingsley. The main theorem says that a countably additive probability measure $P$ on a algebra can be extended to a countability additive ...
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0answers
31 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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0answers
10 views

Regular measure on Borel sets

I am trying to do the following problem: Let $\mu$ be a measure defined on the Borel sets of $\mathbb R^n$ such that $\mu$ takes finite values on the compact sets. Let $\mathcal H$ be the class of ...
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1answer
22 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...
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0answers
36 views

Of which sets does the $\sigma$-algebra generated by the first $n$ one-point sets of $\mathbb{N}$ consist?

Let $n \in \mathbb{N}$ and $\mathcal{E}_n := \{\{1\},\{2\},\dots,\{n\}\}$. The $\sigma$-algebra which is generated by $\mathcal{E}_n$ is defined as follows: $$\sigma(\mathcal{E}_n) := \bigcap ...
2
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2answers
29 views

Why is the zero extension of an $L^p$ function in $L^p$?

Let $u \in L^p(0,1)$. Define $\tilde u:(0,\infty) \to \mathbb{R}$ as the function which equals $u$ on $(0,1)$ and $\tilde u =0$ on $(1,\infty)$. I cannot figure out why this function is measurable. ...
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0answers
15 views

Question on proof of disintegration of measures

In a probabilistic setting: Let $\mu$ be a measure on the product space $S=S_1\times S_2$, both standard Borel, $\mu_1, \mu_2$ the marginal measures. Then there exists a Markov kernel $k$ such ...
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0answers
33 views

Measureable functions and its properties [on hold]

I have two question about measureable functions and its properties and I want some help to solve them $1)$ if $f$ and $g$ are positive measureable functions then $f-g$ is measureable function ? ...
2
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1answer
41 views

Showing $E[X_{n+1}|X_1,…,X_n] = a_0+\Sigma_{k=1}^n a_kX_k$

$X_1,...,X_n,X_{n+1}$ are jointly distributed with a Gaussian distribution. We let $X^* = E[X_{n+1}|X_1,...,X_n]$. Show that there exists constants $a_1,...,a_n,a_{n+1}$ such that $X^* = ...
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0answers
41 views

About an idea in proving Riesz representation theorem for continuous function of compact support.

I tried to prove Riesz representation theorem for continuous linear functional defined on the continuous functions of compact support of a topological locally compact Hausdorff space in this way: 1) ...
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1answer
31 views

If iterated integral is zero then function is zero

We are in Measure & Integration class and were assigned this problem from a chapter on Product Measure & Fubini Theorem: Let $f$ be a real-valued function, integrable with regards to ...
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1answer
42 views

Why every algebra on finite set is a topology

How can I prove that every algebra on finite set is a topology on this set And if the set is infinite how can give me an example algebra but it isn't topology
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1answer
23 views

On showing that if $f_n \to f, g_n \to g$ in $L^p$ then $max(f_n, g_n) \to max(f, g)$ in $L^p$

Let $(f_n)$ and $(g_n)$ be two sequences in $L^p(\Omega)$ with $1 \leq p < \infty$ such that $f_n \to f$ in $L^p(\Omega)$ and $g_n \to g$ in $L^p(\Omega)$. Let $h_n = max(f_n, g_n)$ and $h = max(f, ...
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3answers
46 views

Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue measure

Let $ f : \mathbb R \rightarrow \mathbb R$ be a bounded Lebesgue measurable function such that $\int_a^b f =0$ for all real $a,b.$ Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of ...
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0answers
12 views

Gaussian Measure for Random Matrix

I am doing physics and do not have enough mathematical background. so the question may be trivial, I apologize for that. any help would be highly appreciated. my question is: How does Random Matrix ...
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1answer
16 views

How can I prove that in monotone class

How can I prove that : Let $X$ be nonempty set and $A$ is algebra in $X$ and $A$ is a monotone class , then $A$ is $σ$ Algebra in $X$
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3answers
58 views

How can I compute the following integrals

What is the best way to compute the following integrals $$\int_{0}^{1}\int_{0}^{1}\frac{x^2-y^2}{(x^2+y^2)^2}dydx$$ And $$\int_{0}^{1}\int_{0}^{1}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$$ I know the ...
2
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0answers
33 views

How can I solve like this exercise about measurable function

How can I solve the following exercise Every positive measurable function is limit of increasing sequence of positive simple function How can I prove that I need the proof with explain or how can ...
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1answer
43 views

Basic Properties of Integrals

I am doing a course on measure theory and we are studying integrals and there is a lemma which states: $\int f$ $d\mu$ exists and is greater than $-\infty$ if and only if $\int f^{-} d\mu$ $< ...
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0answers
21 views

There is a Borel measure $ \mu $ on $ [0,1] $ such that $\mu([0,x))=G(0)-G(x) \forall{x\in (0,1]}$

Let $C_c([0,1])=\{f \in C(X): supp (f)$ compact $\}$ and $G:[0,1]\to \mathbb{R}$ be a function monotone nondecreasing continuous and let $\bigwedge:C_c([0,1])\to \mathbb{R}$ define by ...
1
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1answer
16 views

Sigma-algebra: $\sigma(\mathcal{A})=\mathcal{A}$?

Let $\mathcal{A}$ be a $\sigma$-algebra; and let $\sigma(\mathcal{A})$ be also a $\sigma$-algebra generated by $\mathcal{A}$. Because $\sigma(\mathcal{A})$ is the smallest $\sigma$-algebra ...
0
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1answer
19 views

All Borel Sets are measurable

Given an outer measure $\mu^{*}:(X,d) \rightarrow [0, \infty]$, where $X$ is a metric space, will all Borel Sets be $\mu^{*}$-measurable only when $\mu^{*}$ is a metric outer measure? Because if ...