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

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2
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0answers
34 views

quick into 'Function Analysis', ‘Measure Theory’

Can someone suggests some quick introduction document?
1
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1answer
36 views

Lebesgue Measure in ${R}^m$

I am trying to solve the following problem. Problem Statement If $E \subset \mathrm{R}^m$ and $\lambda_m\left(E\right) >1 $, where $\lambda_m$ is the Lebesgue measure on ${R}^m$, then there are ...
5
votes
0answers
71 views
+50

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
3
votes
0answers
25 views

Complete “Macroscopic” Measure

Let $(X,\mathcal{M},\mu)$ be a measure space and let $(X,\overline{\mathcal{M}},\overline{\mu})$ be it's completion. The definition of a complete measure can be interpreted as a complete "microscopic" ...
2
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1answer
38 views

Questions about sigma-algebra

I am learning measure theory this semester. The definition for sigma-algebra is "a collection of sets that is closed under complements and countable unions and intersections." I wonder what does it ...
2
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4answers
83 views

Book recommendation for Measure Theory

What book would you recommend me to read about measure theory and especially the following: Measure and outer meansure, Borel sets, the outer Lebesgue measure. The Cantor set. Properties of ...
0
votes
0answers
21 views

Detail in Definition of Simple Functions

The motivation to this question comes from here, but it's not necessary to see the link to understand my question. In the book of Measure Theory and integration of Folland, we have the following ...
0
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1answer
23 views

Understanding integral equation with locally integrable function

I'm reading a proof in a book and don't understand a crucial step. So we have $f \in L^1_{\text{loc}}(\mathbb R^n)$ and $\phi \in C_c(\mathbb R^n)$. Now $\mu$ is a Radon measure, such that $$ \mu(A) ...
4
votes
1answer
34 views

Measure Theory - Problem with definition about simple functions

I did this question lately and then realized what my mistake was. I got a good help! But looks like only now I understand what was the real problem, to begin with. And this brought me to the same ...
3
votes
0answers
63 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
1
vote
1answer
43 views

Is every simple function on a compact measure space the pointwise limit of continuous functions?

Question: Let $(X,\mu)$ be a measure space and suppose that $X$ is compact. Is every simple measurable function $s:X\to\mathbb{R}$ (i.e. $s(X)$ is a finite set) the pointwise limit of continuous ...
-2
votes
1answer
31 views

Solving this discontinuous integral using Lebesgue

Not a duplicate look at $f(x)$ here! Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is irrational}, & \newline 0 ...
8
votes
5answers
492 views

Evaluating Integrals using Lebesgue Integration

Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is rational}, & \newline 0 \space \text{if} \space x \space \text{is ...
1
vote
1answer
33 views

Basic measure theory question, if there exists a set such that $\mu(E)<\infty$ then $\mu(\emptyset)=0$ - answer check

Basic measure theory question, if there exists a set such that $\mu(E)<\infty$ then $\mu(\emptyset)=0$ If $\mu$ is an extended real valued, non-negative, additive, set function defined on a ring R ...
0
votes
1answer
44 views

Prove that $\int_{[c,d]}|f(x,y)|d\mathcal{L}(y)<\infty$ for $\mathcal{L}$-almost all $x\in [a,b]$.

Suppose $f(x,y)$ is a Borel function on $\mathbb{R}^2$ which is in the $L^2$-space with respect to the $\mathcal{L}\times\mathcal{L}$. Prove the following: Given any finite rectangle ...
1
vote
1answer
35 views

Why is this 2 here (I believe I have shown it without it) - integration (Riemann integral)

I am looking at Proposition 1.3 (on page 3, how embarrassing!) The line I dispute is $I_\mathcal{P}(f)\le I_{\mathcal{P}_1}(f)+\frac{2M}{k}l(I)$ I see no need at all for the 2! Logic: ...
2
votes
0answers
70 views

Exercise in Probability/Measure Theory

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let also $A_{n,j}\in\mathcal{F},n\in\mathbb{N}_0,j\in\{1,2,3,...,2^n\}$, be such that for all ...
-3
votes
0answers
25 views

why we need semi algebra and algebra for explaining sigma algebra? [closed]

In measure theory introduction why we need semi algebra and algebra to explain about sigma algebra ? what is it's main purpose?
2
votes
2answers
26 views

Help with a Royden exercise of measure

I'm solving the exercise 12, of section 4 The General Lebesgue Integral from the Royden's book Real Analysis 3rd edition: Let $g$ be an integrable function on a set $E$ and suppose that $(f_n)$ is a ...
1
vote
0answers
21 views

Inequality almost everywhere [closed]

Let $(X, F, u)$ a measure space and $(f_n)$ a sequence of measurable functions. Suppose that $ \left| {f_n} \right| \leq {g}$ in u_ctp and $g$ is integrable. Assuming the dominated convergence theorem ...
1
vote
1answer
38 views

I don't understand part of a proof involving sigma algebra

It must be pretty trivial but I don't understand one part for the reverse containment relation, I don't understand why A* contains A` and why we A* is sigma field on $\Omega^{*}$ notice here ...
7
votes
1answer
128 views

Lebesgue point of density on $[0,1]$ and Dynkin's theorem

The problem defines a density point $x\in[0,1]$ for a Borel set $A\subset [0,1]$ if $$ \lim_{\varepsilon \rightarrow 0^+} \frac{\mu([x-\varepsilon,x+\varepsilon]\cap A)}{2\varepsilon}=1.$$Denote all ...
1
vote
0answers
33 views

Can I apply measure theory in non-mathematics fields?

I am working in a field where researches try to get insight about a complex process. I will give an example to demonstrate this. Let's say, we are attempting to get the most efficient and cost ...
1
vote
2answers
35 views

Question about sigma-algebra's

I currently reading a measure theory book but I have something I don't quite understand why is sigma-algebra iff its both a $\lambda$-system and a $\pi$-system. I am having troubles understanding why ...
2
votes
1answer
58 views

Prove existence of borel set related to the function $f(x)=2x \mod 1$

Let $I=[0,1)$ and $f(x)=2x \mod 1$. Prove that for every $\epsilon>0$ there is $E\subset I$ borel set s.a $m(I/E)<\epsilon$ and $\lim_{N\to\infty} \sup \{|\frac{1}{N}\sum_{j=0}^{N-1} ...
2
votes
1answer
24 views

Condition in a theorem in Probability theory.

I passed by a simple theorem in Probability theory , yet it really bugs me that I think that 1 condition in the hypothesis is not necessary. After checking the proof for many times, I still can't ...
1
vote
1answer
38 views

Measurable functions and the Cauchy condition

Suppose we have a measure space $(\Omega, \Sigma, \mu)$ and measurable functions $f_n \colon \Omega \to \mathbb R$. Is it true, that if the sequence $f_n$ is convergent in measure, then it is ...
0
votes
1answer
30 views

Computing the $\sigma$-algebra $\sigma ( \{ \{ a \}:a \in \mathbb{Q} \})$ on $\Bbb R$

I found an interesting question in a book. Question: Compute the $\sigma$-algebra $\sigma ( \{ \{ a \}:a \in \mathbb{Q} \})$ on $\mathbb{R}$. What is interesting to me is that the problem comes ...
4
votes
2answers
73 views

How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam's original argument about measure theory and measurable cardinals. Here is the result I am looking for: ...
2
votes
2answers
45 views

About a solution of Measure Theory and Integration

The problem is from Folland's book of Measure Theory and Integration. In this problem, $(X,\mathcal{M}, \mu)$ is the measure space and $L^+$ is the space of measurable functions $f:X\to[0,\infty]$. ...
1
vote
1answer
39 views

Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
1
vote
0answers
65 views

Is the set of all Lebesgue-measurable sets measurable?

I consider the set $X^p=\{A\subseteq[0,1]|A$ Lebesgue-measurable and $\lambda(A)=p\}$ for a $p\in (0,1]$. My objective is to construct a random variable with values in $X^p$. Therefore I need to know ...
3
votes
1answer
130 views

On a proof of Riesz-Fischer Theorem

Questions : [See below for context.] $\rm\color{#c00}{a)}$ First, is the proof presented below $100$ % correct ? $\rm\color{#c00}{b)}$ How would one justify the LHS of $(2)$ ? Are my ...
1
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0answers
7 views

Proof regarding cumulative distribution functions…

Here is the theorem: Theorem The following statements are equivalent: a. The random variables $X,Y$ are identically distributed b. $F_X(x)=F_Y(x)$ $\forall x$ In our book, they ...
1
vote
2answers
63 views

Measure Theory-Borel sets-Lebesgue integral-Monotonce Convergence Theorem question

I am preparing for an exam in measure theory and probabilities and the question below is from a previous exam in this course. I have tried to answer it, though I miss certain key points in my ...
2
votes
1answer
55 views

Lebesgue measure in one dimension

Let $A$ be Lebesgue measurable and $0<\lambda(A)<\infty$. Let $\alpha\in(0,1)$. Prove that there exists an open interval $P$ such that: $$\lambda(A\cap P)\leq\alpha\lambda(P)$$ I found a proof ...
1
vote
2answers
23 views

Suggested measure theory books for certain exercises

I was wondering if anyone knows books with difficult exercises of the theorems of monotone and dominated convergence and if the motto of Fatou possible. I use Bartle but it does not have many ...
1
vote
2answers
47 views

Let $E\subseteq\mathbb{R}$ be a borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Is $m(f(E))=0$?

Let $E\subseteq\mathbb{R}$ be a Borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Is $m(f(E))=0$? I think it is true, but I do not know how to prove it. The only think I have got is that, if ...
1
vote
1answer
38 views

Fubini-Study measure on a product

I read that if $\Omega_2$ is the Fubini-Study form on $\mathbb{P}_1\times\mathbb{P}_1$, and $\Omega$ the Fubini-Study form on $\mathbb{P}_1$, then for all $(x,y)\in\mathbb{P}_1\times\mathbb{P}_1$, one ...
1
vote
1answer
38 views

Conditional independence of sigma-algebras

If ${\mathcal{H}_1}$ and ${\mathcal{H}_2}$ are conditionally independent given $\mathcal{G} \subseteq {\mathcal{H}_2}$, are they conditionally independent given $\mathcal{F}$ such that $\mathcal{G} ...
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vote
0answers
16 views

Let $\mu$ be a finite borel measure & $\nu$ a complex borel measure on $\mathbb R$ if $\mu (I)\ge|\nu (I)|$ then $\mu (E)\ge|\nu| (E)$

Let $\mu$ be a finite Borel measure on $\mathbb R$ and $\nu$ a complex borel measure on $\mathbb R$ if $\mu (I)\ge|\nu (I)|$ for every interval in $\mathbb R$ then for all $E\in\cal B_{\mathbb R}$ we ...
0
votes
1answer
30 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...
1
vote
2answers
60 views

Regularity of Dirac measure on Baire sets

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
4
votes
1answer
72 views

Existence of regular Borel measure

Let $X$ be a $\sigma$-compact and locally compact space, and let $\Lambda:C(X)\rightarrow \mathbb{C}$ be a linear functional such that $\Lambda(f)\ge0$ if $f\ge0$. How to show that exist exactly one ...
2
votes
1answer
30 views

Existence of a function

Let $D=\{z\in\mathbb{C}:|z|<1\}$ How can one show that there exist function $f:[0,1]\times D \rightarrow \mathbb{C}$ satisfying the following properties: (i) $f(\cdot,z)$ is continous on $[0,1]$ ...
0
votes
1answer
41 views

Divergence Theorem/Integration by Parts on Unbounded Domains

Are there any formulations of the Divergence Theorem or integration by parts formulae that apply to unbounded domains?
3
votes
1answer
39 views

show $\lim_{n\rightarrow\infty}\int_X|f_n-f|d\mu=0$

I want to show $\lim_{n\rightarrow\infty}\int_X|f_n-f|d\mu=0$ for integrable functions $f_n,f:X\rightarrow [0,\infty)$, $f_n\rightarrow f$ pointwise a.e. and $\int_Xf_nd\mu\rightarrow \int_Xfd\mu$. ...
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votes
0answers
17 views

Function Varied Over Indicator Functions is Zero on a Generating $\pi$-System

Let $f$ be an integrable function on a measure space ($E,\mathcal{E},\mu)$. Suppose that, for some $\pi$-system $\mathcal A$ containing $E$ and generating $\mathcal E$, ie $\sigma(\mathcal A) = ...
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vote
0answers
24 views

Why $m(E\backslash E_x)\leq m(E\backslash E')+m(E'\backslash E'_x)+m(E'_x\backslash E_x)\leq …$

Let denote $E_x=E+x$. I have to prove that $$\lim_{x\to 0}m(E\backslash E_x)=0$$ for $E\subset \mathbb R$ measurable s.t. $m(E)<\infty $. I know that there exist $E'=\bigcup_{i=1}^N Q_i$ where the ...
1
vote
3answers
27 views

Proving Lebesgue measurability of Dirichlet-like functions

Dirichlet function $D:[0;1]\to\mathbb{R}$ is defined by $$ D(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \not\in \mathbb{Q} \end{cases}$$ We say that a function ...