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

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Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
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0answers
15 views

The relationship between outer measures and smallest coverings

Recall that if $\mathcal{A}\subset \mathcal{P}(X)$ is an algebra and $\mu_{0}:\mathcal{A} \to [0,\infty]$ is a premeasure on $\mathcal{A}$ then we can define the outer measure $\mu^{*}$ for any set ...
0
votes
1answer
34 views

Understanding Cohn's Radon-Nikodym proof from his book on measure theory

The part of the proof which I don't get is $$\nu(A)=\int_{A} g\ \mathsf d\mu$$ where $g$ is Radon-Nikodym derivative. He has a set of functions for which $$\int_{A} f\ \mathsf dx \le \nu(A) ,$$ he has ...
2
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0answers
19 views

Generalization of the Vitali-Hahn-Saks Theorem

Is there a generalization of the Vitali-Hahn-Saks Theorem for nets of measures? I do not find any related literature.
2
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0answers
24 views

Is there a programmatic way to calculate cascaded sigma functions?

Let my format be sigma(function,from,to) = f(n) for example sigma(sigma(1 , j = 1 , j = i) , i = 1 , i = n) = (n^2)/2 ...
2
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0answers
10 views

How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
1
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0answers
23 views

Expectation of a continuous function

Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and ...
4
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1answer
58 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...
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1answer
13 views

Under the Borel measure associated to the Cantor function each of the intervals remaining has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
1
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1answer
19 views

Question on mutual singularity and absolute continuity of complex measures

I was presented these two somewhat similar questions from Folland's real analysis (second edition) dealing with complex measures and their mutual singularity and absolute continuity. They are 3.19 and ...
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0answers
15 views

Behaviour of Fourier Inverse Transform after non-linear modulation

Suppose $\phi$ is a continuous nowhere differentiable function. $g$ some function in Schwartz space such that $\hat{g}$ has compact support. Define $f(x) = \int_{-\infty}^\infty ...
1
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1answer
30 views

A subset on which a measure is strictly positive

There is a "cake" which is represented by the interval $[0,1]$. There is a non-atomic value measure $V$ defined on the cake. I would like to define an algorithm for dividing the cake between two ...
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1answer
45 views

Find Lebesgue measure of $\limsup A_n \cap B_n$ if $m(\limsup A_n)=m(\liminf B_n)=1$,

Let $m$ be the lebesgue measure on $X=[0,1]$. if $m(\limsup\limits_{n\rightarrow{\infty}} {A_n})=1$ and $m(\liminf\limits_{n\rightarrow{\infty}} {B_n})=1$, prove that ...
3
votes
1answer
24 views

Difference between a measure and a premeasure

I am new to measure theory and am wondering: Is the only difference between a measure and a premeasure the fact that measures are defined on $\sigma$-algebras and premeasures are defined only on ...
2
votes
1answer
18 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
9
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3answers
779 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
2
votes
2answers
33 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
0
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1answer
30 views

$\limsup = \liminf$ of sequence of Sets

This problem was on my in-class final for a measure theory course I took in the fall, and now I am studying for my qualifying exam so I am trying to figure this one out: Suppose ...
1
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1answer
20 views

Does convergence in $L_p$ imply convergence of quotients in $L_p$

Take the measure space to be $\mathbb{R}$ with Borel $\sigma$-algebra and Lebesgue measure (Although just thinking in terms of a general measure space probably works for this problem.) Question: True ...
1
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1answer
50 views

Definition of upper integral

Answering this question it occurred to me that the OP's definition of integral is unsatisfactory in the following sense. He defines it using the usual Lebesgue integral. I think it would be far more ...
1
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2answers
33 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
9
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2answers
112 views

Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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0answers
43 views

A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
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1answer
28 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
1
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1answer
24 views

What is the countable product sigma algebra of powersets of a countable set $E$? The powerset of the space of all sequences in $E$ or not?

Let $E$ be a countable set with power set $\mathcal{P}(E)$. $(E,\mathcal{P}(E))$ is a measurable space. Let $E^{\mathbb{N}}$ be the space of sequences in $E$ and $\mathcal{P}(E)^{\mathbb{N}}$ the ...
0
votes
1answer
15 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
5
votes
2answers
101 views

Explicit construction of a nonmeasurable set, where only the proof of correctness uses choice?

By Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence ...
1
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0answers
20 views

Specific problem on Radon measures from Folland's real analysis on $ C_0(X) $

Hello all I am trying to understand the concept of $ C_0(X) $ within the concept of Radon measures as presented in Folland's real analysis chapter 7, so far so good right until I came across problem ...
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2answers
68 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
2
votes
2answers
43 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
3
votes
1answer
29 views

Weak compactness of a set of translates in $C_0(\mathbb{R})$

Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$. I want to show that $A$ is relatively compact in the ...
0
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2answers
31 views

Totally ordered $\sigma$-algebras

I know that every $\sigma$-algebra is partially ordered with respect to the inclusion operator $\subset$. However, it seems as though every $\sigma$-algebra should be totally ordered with respect to ...
0
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2answers
37 views

Product Integral: Integrability

Given measure spaces $X$ and $Y$. Then it holds: $$\int_Y\int_X|\eta(x,y)|\mathrm{d}\mu(x)\mathrm{d}\nu(y)<\infty\implies\int_X|\eta(x,y)|\mathrm{d}\mu(x)<\infty\quad(y\in Y)$$ Can this ...
1
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0answers
32 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
2
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1answer
32 views

Compact $K\subset A$ such that $\lambda(K) = \lambda(A) / 2$

Let $A\subset \mathbb{R}$ be a (Lebesgue) measurable set of finite measure. Using the fact that the function $f:\mathbb{R}\rightarrow \mathbb{R}$, $$f(x)=\lambda(A\cap [-x,x]) $$ is continuous, we ...
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0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
0
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1answer
25 views

Trace $\sigma$-algebra and measurable envelope

I'm stuck on a problem from Cohn's book. Let $(X,\mathscr{A})$ a measurable space, and let $C$ be a subset of $X$. Let $\mathscr{A}_C$ be the trace of $\mathscr{A}$ on $C$, that is all the ...
2
votes
0answers
36 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
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0answers
37 views

Question on Radon measure's Lebesgue decomposition

Hi all seeing as how people were so nice to me and my experience was a success I though perhaps it was safe to try and ask this as well on Radon measures (also same class) I am given a $ ...
4
votes
1answer
44 views

Question on Radon measures from Folland's Real Analysis

Greetings my mathematical friends. I am taking a summer class on measures and the theory of real analysis, and I was given the following question from Folland's Real Analysis Second Edition Chapter 7 ...
2
votes
1answer
43 views

On Wikipedia's article Carathéodory's extension theorem

Wikipedia's article(https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem) says: Let $R$ be a ring on $\Omega$ and $\mu: R \rightarrow [0, +\infty]$ be a pre-measure on $R$. The ...
3
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1answer
24 views

Coincidence of two $\tau$-additive measures

I'm struggling to prove the following Lemma from V.I. Bogachev, Measure Theory 2: Let two $\tau$-additive measures $\mu$ and $\nu$ on a topological space $X$ coincide on all sets from some class ...
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0answers
36 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
0
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1answer
17 views

Help verify my proof of “countable additivity holds for Lebesgue measure when every pair of sets disjoint a.e.”

This is the question, where $|*|$ denotes Lebesgue measure. Let ${E_j}$ be a sequence of Lebesgue measurable sets in $\Bbb{R}^n$ st. $|E_j⋂E_i |=0$ for $j≠i$ (i.e. they are pairwise disjoint ...
1
vote
1answer
61 views

How to prove a set contains no rational numbers?

Let $E\subseteq \Bbb R$ be a set of Lebesgue measure zero. Show that there exists $a \in \Bbb R$ such that the set $$E+a :=\{x+a:x\in E\}$$ contain no rational numbers. I tried to use there is a ...
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1answer
59 views

Question 7.7 in measure theory on Radon measure from Folland's Real Analysis Second Edition

Hello all I was presented with this question from Folland's real analysis second edition on Radon measures which I am stuck on and so would really appreciate the help on. I m a novice in Radon ...
1
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2answers
28 views

Number of possible unions of a countable number of sets

If $\{ A_{n} \}_{n=1}^{\infty}$ is a countable sequence of distinct sets, then is the number of possible distinct unions between any two or more of the sets in the sequence uncountable? I would like a ...
0
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1answer
30 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
3
votes
3answers
58 views

Help with $\int_0^\infty {\frac{{\sin t}}{{{e^t} - x}}dt} = \sum\limits_{n = 1}^\infty {\frac{{{x^{n - 1}}}}{{{n^2} + 1}}} $

The question is to show $\int_0^\infty {\frac{{\sin t}}{{{e^t} - x}}dt} = \sum\limits_{n = 1}^\infty {\frac{{{x^{n - 1}}}}{{{n^2} + 1}}} $ for $-1<x<1$. The integration is a Lebesgue ...
0
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0answers
20 views

An example of Lebesgue measurable set but not Borel measurable besides the “subset of Cantor set” example. [duplicate]

The question is to give and example of Lebesgue measurable set but not Borel measurable. I know there exists subset of Cantor set that is not Borel measurable, since the cardinality of all Borel sets ...