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

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5
votes
2answers
106 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
vote
0answers
21 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 ...
1
vote
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
45 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 ...
4
votes
1answer
34 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
votes
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
votes
2answers
38 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
vote
0answers
33 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
votes
1answer
33 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 ...
0
votes
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
votes
1answer
31 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
37 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 ...
5
votes
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
45 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
votes
1answer
26 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 ...
0
votes
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
votes
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
62 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 ...
7
votes
1answer
60 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
vote
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
votes
1answer
32 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
62 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
votes
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 ...
0
votes
1answer
30 views

Help with a Lebesgue integration problem.

The question is the following, Let $f:\Bbb{R}\rightarrow \Bbb{R}$ be a Lebesgue integrable function. Show that $\mathop {\lim }\limits_{t \to \infty } \int_\Bbb{R} {f(x)\cos (xt)dx} = \mathop ...
4
votes
3answers
57 views

Change of variable for a limit inside Lebesgue integration?

To calculate $\lim\limits_{n \to \infty } \int_A \cos (nx) \, dx $ where $A$ is a compact set, say $[0,1]$, the objective is to show the integral $\rightarrow 0$. My question is can I first exchange ...
1
vote
1answer
66 views

Measure converges to zero

I'm trying solving the following problem: Let $f:[0,1]\to \Bbb{R}$ be a measurable question such that $f(x)>0$ a.e. Let $\{E_k\}_{k=1}^\infty\subset [0,1]$, a sequence of set such that ...
1
vote
1answer
41 views

Conditional Radon Nikodym

I am having some trouble conceptualizing and calculating a conditional RN derivative. When using this definition: I can see that if $\mathbb{Q} \ll \mathbb{P}$: $$\mathbb{E}_\mathbb{Q}(g) = \int ...
1
vote
0answers
39 views

Convergence of argmax of poisson point process under a continuous map

This might just be a simple measure theory question, see the end remark, and apologies if that's the case. Here's the setup: Let $\mu_n$ be a sequence of Poisson point processes on a Euclidean space ...
1
vote
0answers
35 views

Definition of an infinite $\sigma$-algebra

What is the formal definition of an infinite $\sigma$-algebra? I cannot find this definition anywhere! But based on the context I have read it in I think it is either: (A) A $\sigma$-algebra ...
2
votes
1answer
23 views

Relating Integration by Substitution to Change of Variables Theorem

I'm having trouble relating the change of variables theorem from measure theory to the integration by substitution formula taught in Calculus. I've always thought they were basically saying the same ...
0
votes
0answers
16 views

The weighted Signed measure

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\mu$ be a finite signed measure. Then we know that, for $\varphi\in C_c^\infty$, $$ \sup_{\|\varphi\|_\infty\leq ...
3
votes
0answers
62 views

Extension of measure from $\sigma$-ring to $\sigma$-algebra

Halmos developed measure theory based on $\sigma$-rings. Nowadays measure theory is based on $\sigma$-algebras. I would like to know how to bridge the two theories if possible. Namely, let $(X, ...
1
vote
2answers
144 views

Lang's treatment of product of Radon measures

Let $X$ be a locally compact Hausdorff space. We denote by $\mathcal B(X)$ the $\sigma$-algebra of Borel sets of $X$. A positive Radon measure $\mu$ on $X$ is a measure defined on $\mathcal B(X)$ with ...
1
vote
1answer
28 views

Sequences of functions that converge uniformly, or pointwise, but not in $L^1.$

I'm reading the book Real Analysis of Folland. When I reached chapter 2 about the different modes of convergence, there's an example Folland gave that confused me: The 2 function sequences: ...
2
votes
1answer
31 views

Show if $f$ is a measurable function, then for any Borel set $B$, $f^{-1}(B)$ is measurable.

How to formally show if $f$ is a Lebesgue measurable function, then for any Borel set $B$, $f^{-1}(B)$ is measurable. The theorem I know is that if $f$ is Lebesgue measurable, then for any open ...
1
vote
1answer
36 views

$f\nu_{}=\big.m\big|_{[0,1]}$ where $m$ is the Lebesgue measurable.

If consider the map $f:2^{\omega}\to [0,1]$ given $f(x)=\sum_{i=0}^{n}x(i)2^{-i-1}$. Let $\nu_{}$ be the Haar measurable on $2^{\omega}$. Then $f\nu_{}=\big.m\big|_{[0,1]}$ where $m$ is the Lebesgue ...
1
vote
1answer
53 views

Abstract enunciation of the Good Set Principle in measure theory

I am struggling with the Good Set Principle in Measure Theory, so is this rephrasing in the most abstract way ultimately correct? Good Set Principle Let $(X, \Sigma)$ be a measurable space. ...
1
vote
2answers
26 views

Can we find a unique probability Borel measurable $\mu$ on $\mathcal{C}$ with $\mu(N_s)=\phi(s)$

If $\phi:2^{<\mathbb{N}}\to [0,1]$ satisfies $\phi(\emptyset)=1$ and $\phi(s)=\phi(s^{\widehat{}}0)+\phi(s^{\widehat{}}1)$ for all $s \in 2^{<\mathbb{N}}$. Can we find a unique probability ...
1
vote
0answers
17 views

Proof Check on Change of Variables Result

Let $g: I \to \mathbb{R}$ be strictly increasing with continuous derivative on an open interval $I \subset \mathbb{R}$. Let $\mu$ be the measure on $(I, \mathcal{B}(I))$ with density $g^\prime$ ...
3
votes
2answers
39 views

Help with a variant of a theorem

The original theorem states that Let $ϕ$ be continuous on $\Bbb{R}$, let $f$ be finite on $Ω$ a.e., then $ϕ∘f$ is measurable if $f$ is measurable. Now is it true for a slightly different ...
0
votes
1answer
33 views

If $f,g$ are measurable functions, $g \neq 0$ a.e , show $f/g$ is measurable.

If $f,g$ are measurable functions, $g \neq 0$ a.e , show $f/g$ is measurable. I know how to prove $fg$ is measurable. Since $x^2$ is a continuous function on $\Bbb{R}$, and $f+g$, $f-g$ are ...
3
votes
2answers
119 views
+400

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
3
votes
4answers
116 views

$f,g$ are both measurable on a set $ \Omega $, can $ \{ x\in \Omega: f(x)=g(x) \} $ be non-measurable?

Suppose $f,g$ are both measurable on a set $ \Omega $, can $ \{ x\in \Omega: f(x)=g(x) \} $ be non-measurable? My attempt: Let $\Omega$ be an open interval in $\Bbb{R}$, then it has a non-measurable ...
0
votes
1answer
26 views

Closeness of a number to mean.

Let's say I am given mean $\mu$ and deviation $\sigma$ of a set of numbers. I am now given $x$, a real number. Depending on how close $x$ is to $\mu$, I need a measure starting from 100 going down ...
1
vote
1answer
35 views

Sequence of integrals of positive function

Let $f(x)$ be a function positive almost everywhere on $X$. Let $A_n$ be a sequence of subsets of $X$ such that $m(A_n) > c> 0$ for all $n$, where $c$ is some constant, and $m$ denotes the ...
1
vote
2answers
28 views

Definition of the product $\sigma$-algebra

The following is the definition of the product $\sigma$-algebra given in Gerald Folland's Real Analysis: Modern Techniques and Their Applications (pg. 22) (note that $\mathcal{M}(X)$ denotes the ...
2
votes
2answers
34 views

Absolutely continuous measure

If I have a measure $\mu$ on $[0,1]$ and if I know that $\int_{[0,1]}Gd\mu\leq\int_0^1|G(r)|dr\quad \forall G\in C[0,1]$ this implies that the measure $\mu$ is absolutely continuous with respect the ...
0
votes
0answers
17 views

Uniformly continuous unitary representations.

Let $U$ be a unitary rep. of $\mathbb{R}^d$ on a separable Hilbert space $H$, and $H\cong\oplus L^2_{\mu_v}(\mathbb{R}^d)$ be the spectral decomposition (according to the spectral theorem for these ...
1
vote
1answer
20 views

uniform lower bound for integrals of almost everywhere positive function

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, and let $f$ be a function defined on $\Omega$ which is positive almost everywhere. Let $c$ be a fixed positive constant such that $c < ...