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

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What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
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1answer
66 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
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1answer
9 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 ...
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0answers
27 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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0answers
16 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 ...
2
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0answers
33 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 ...
2
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1answer
42 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

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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1answer
31 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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27 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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2answers
107 views

Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$

Suppose $f,f_n$ are measurable and uniformly bounded on $[a,b]$. Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$ Attempt: We note that since $f$ and $f_n$ are bounded and are ...
1
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1answer
81 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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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 ...
2
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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
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
38 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 ...
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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 ...
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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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1answer
324 views

Finding two functions (density) $g,f$ satisfying some conditions

Is there a clever way to find two density functions, $f$ and $g$, that satisfy the following conditions? $$\begin{align*} ...
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0answers
26 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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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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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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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 ...
1
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1answer
72 views

Is it true that $ \int_{[1,\infty)} f_n\to \int_{[1,\infty)} f$?

Can you please help me solve this on measure theory? My TA did not go over this. He said we are not going over this but you can do this if you want. Can someone please explain to me? Thanks. Suppose ...
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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
15 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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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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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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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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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!
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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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2answers
128 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
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0answers
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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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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1answer
42 views

Question about working area of Vitali cover

$\text Carothers' Real Analysis$ defines his Vitali cover with no introduction which made me confused a lot. Here is the definition of a Vitali Cover: I'm not sure when does a set have its Vitali ...
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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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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
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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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 ? ...
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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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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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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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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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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^* = ...
5
votes
1answer
553 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
1
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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) ...