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

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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
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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
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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
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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 ...
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1answer
45 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 ...
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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
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0answers
72 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 ...
2
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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 ...
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1answer
40 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
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1answer
143 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 ...
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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
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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
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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
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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
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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
75 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: ...
3
votes
2answers
49 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
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1answer
42 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 ...
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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
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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 ...
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0answers
8 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
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2answers
65 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
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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 ...
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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
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2answers
48 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
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1answer
40 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
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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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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
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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
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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
73 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 $[0,1]$ ...
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1answer
42 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
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1answer
41 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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0answers
18 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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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
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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 ...
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0answers
30 views

Convergence in finite-dimensional distributions of stochastic processes and random measures

Motivation: Let $(\Omega, \mathscr{A}, P)$ be a probability space and consider the measurable spaces $D := D[0,\infty)$ of cadlag functions and $M := M[0,\infty)$ of Radon measures on $[0,\infty)$. ...
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0answers
16 views

exists borel subset such that for every nonempty open dyadic interval [duplicate]

How would I go about showing that there exists a Borel subset $B \subset [0, 1]$ such that for every nonempty open dyadic interval $J \subset [0, 1]$,$$0 < \mu(B \cap J) < \mu(J),$$where $\mu$ ...
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2answers
26 views

Part of Fubini's Theorem with almost everywhere

Here is the statement: ($\bf{Tonelli}$) If $f\in L^+(X,Y)$, then $\displaystyle g:x\mapsto\int_Yf_xd\nu$ is $\mathcal{M}$-measurable,\ $\displaystyle h:y\mapsto \int_Xf^yd\mu$ is ...
1
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1answer
29 views

Limit of density points is a density point? [closed]

Let $\lambda(A)>0$ for some $A \subset \mathbb{R}^n$ and let $x_n \in A$ be a sequence of Lebesgue density points of $A$ with $x_n \to x \in int(closure(A))$. Must $x$ be a density point as well?
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1answer
31 views

Integration over a conditional cumulative density function?

In his text "Mathematical Statistics: A Decision Theoretic Approach," Ferguson describes a way to define the expected value as a Rieman-Stieltjes integral. Roughly, he says that we can define the ...
3
votes
1answer
31 views

Existence of a probability space

Let us assume that we are given a family of Markov chains $(X^\alpha_t)_{t\geq0}$ in continuous time. Kolmogorov's result ensures that for each $\alpha \in I$ there exists a probability space ...
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1answer
24 views

Determine $\sigma(A)$ for $A=\{\{x\}|x\in\Omega\}$

We had the following exercise in our exam and I am interested in a solution. Let $\Omega$ be a set and $A=\{\{x\}\mid x\in\Omega\}$. Determine $\sigma(A)$. I think $\sigma(A)=\{D\subset \Omega: ...
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1answer
17 views

Simple functions and mesurabilty

I have the question to the following: Let $(X, \mathcal{A})$ be a mesrurable space and $f: X \rightarrow [0,\infty]$. Show that f is measurable iff there are $a_n \in [0,\infty]$ and $A_n\in ...
1
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1answer
31 views

Question about Royden's proof about countable subaddivity of lebesuge outer measure

I have a question about the following proof by Royden. What I do not understand is the part where it says $\Sigma \Sigma l(I_{k,i}) < \Sigma m^*(E_k)+\epsilon / 2^k$ Why is there a strict ...
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1answer
24 views

Area of set-difference of special sets

In a topological space, call a set $X$ special if it is equal to the closure of its own interior (is there a standard term for this?): $$X = \text{Cl}[\text{Int}[X]]$$ Let $X$ and $Y$ be two special ...
1
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1answer
93 views

Existence of T-Vitali sets…

As I understand it Turing degrees are defined as the equivalence classes of sets under the equivalence relation defined by $x \sim y$ iff $x$ is Turing reducible to $y$ and $y$ is Turing reducible to ...
1
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0answers
32 views

Borel sigma algebra and an extra point

I have a question about Borel sigma algebra on a topological space. Let $E$ be a Hausdorff topological space and $\mathcal{B}(E)$ denotes its Borel sigma algebra. We adjoin an extra point $\Delta$ ...