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

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A simple implication of an approximation theorem by Komlós, Major and Tusnády

I have been reading through the PhD thesis of Professor Aue on change point analysis based on invariance principles. There's a particular argument I have not been able to follow: Let ...
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1answer
17 views

Proof that outer measure of interval equals length, why use Heine-Borel?

Define by $$ m^*(A) := \inf\left\{ \sum_i |I| : A \subseteq \bigcup_i I_i \right\} $$ the outer measure of some set $A \subseteq \mathbb R$. Then we have $m^*(I) = |I|$ for each interval (open, ...
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0answers
33 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ if I know the density of $(X_2,X_3)$? The ...
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2answers
58 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...
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0answers
20 views

Show that a collection of finite unions of sets of the form $(a,b]\cap \mathbb{Q}$ is an algebra

The following is a question from Folland's Real Analysis: Modern Techniques and their Applications. (Question 23 page 32) Let $\mathcal{A}$ be the collection of finite unions of sets of the form ...
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1answer
31 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
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1answer
38 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
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26 views

Why do we need a Borel function in order to use this lemma?

Im trying to understand a proof for differentiably a.e for functions $F$ given by $$F(x)= \int_{-\infty}^{x}f\ \mathsf dt$$ for $f$ Lebesgue measurable and $L^{1}$. He defines a finite Borel measure ...
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1answer
15 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
3
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1answer
80 views

Help with a proof regarding simple functions.

The question is If $f>g≥0$, then there exists non-negative measurable simple functions $f_k↗f$ s.t. $f_k≥g$ for all $k$. My attempt. Using a theorem in my text book For every ...
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2answers
49 views

Incomplete measure space that is not sigma-finite

I am looking for an example of an incomplete measure space with a measure that is not sigma-finite. All the measures which are not sigma-finite which I have come across so far are the following: ...
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0answers
102 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
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23 views

Maximal cake-cutting

Alice and George divide a cake between them. The cake is a 1-dimensional interval and both players value the entire cake as 1. The valuations of the players are represented by non-atomic measures on ...
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1answer
43 views

Does simply-connected imply measureable?

The famous examples of non-measureable sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space ...
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2answers
38 views

The exterior measure of a closed cube is equal to his volume.

The proof goes like that: Let $Q\subset \mathbb R^d$ a closed cube of $\mathbb R^d$. Since $Q$ covers itself, we must have $m*(Q)\leq Q$. Therefore, it suffice to prove the reverse equality. Let ...
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0answers
37 views

Does a choice of measure on $\mathfrak{g}$ induce a measure on $G$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. One can put a (left) Haar measure $\mu$ on $G$ and a Lebesgue measure $\lambda$ on $\mathfrak{g}$ which are both unique up to constants. My ...
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1answer
23 views

“Occupation time” nonlinear functional measurable?

My question is for which functions $f$ the following nonlinear functional $f\rightarrow\int \mathbf{1}_B(f(x))dx$ is Borel measurable; $B\in\mathcal{B}(\mathbb{R})$ and $\mathbf{1}_B(.)$ is a ...
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1answer
24 views

Does almost everywhere differentiablty imply existence of weak derivitive?

Does almost everywhere differentiablty imply existence of weak derivitive? What about the converse? If not in general maybe on compacts?
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1answer
41 views

Is σ-finiteness necessary for the “in measure” version of the dominated convergence theorem to hold?

Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$ Now, this may be already solved in the ...
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27 views

Measurability of functions with respect to $F(x,y) =xy.$

We know $F$ maps $[0,1]^2$ into $[0,1]$ and this induces a $\sigma$-algebra $\mathcal{M}=\{F^{-1}(B): B \text{ is Borel subset of }[0,1] \}$ on $[0,1]^2$. How can we describe the ...
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2answers
77 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
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0answers
31 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
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1answer
15 views

A sigma-algebra 'generated by closure under the operation of countable union'

I'm reading a paper which contains the following: We have the following notation. If R is a partition of [Omega], and w [in] [Omega], we denote by R(w) the unique member of R containing w. Also, ...
3
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0answers
23 views

Associativity of product measures

Suppose we have measure spaces $(X_i, M_i, \mu_i), i=1, 2,3$, that are complete and $\sigma$-finite. I learned how to form a product measure from two measure spaces, but I wasn't so sure about product ...
4
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1answer
36 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 ...
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2answers
57 views

Does construction of infinite product measure require axiom of choice?

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...
0
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1answer
33 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
0
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1answer
26 views

Question about the definition of convergence in measure.

In my text book convergence in measure ${{f}_{k}}\mathop \to \limits^{m} {f}$ is defined as "$\forall \epsilon> 0$ we have $\mathop {\lim }\limits_{k \to \infty } |\{ x \in \Omega :\left| ...
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2answers
33 views

Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Let $f_n$ is a sequence of measurable functions on a measure space $(X,\mathcal{B},m)$ converging pointwise to a function $f$. Suppose that $f_n$ is integrable for all $n$ and ...
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1answer
1k views

Closure, Interior, and Boundary of Jordan Measurable Sets.

This question has a number of parts. Let $E\subset\mathbb{R}^{d}$ be a bounded subset. (1) Show that $m^{\star,(J)}(E)=m^{\star,(J)}(\bar{E})$ (closure) (2) Show that ...
3
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2answers
133 views

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 ...
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2answers
1k views

Uniform integrability and tightness.

Definition: Let $(X,M,\mu)$ be a measure space and $\{f_n\}$ a sequence of measurable functions on $x$ that are integrable. Then $\{f_n\}$ is uniformly integrable if for every $\epsilon >0$, there ...
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2answers
43 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
3
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1answer
57 views

Existence of Lebesgue measure on real line proof help

I am reading a proof of the existence of Lebesgue measure and am struggling to understand one part. I will first get you up to where I am in the proof. We define for a set written as a finite ...
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2answers
84 views

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 ...
2
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2answers
58 views

Measures on $\mathbb{R}$ that are not translation invariant

I am looking for examples of measures on $\mathbb{R}$ which are not translation invariant. The only one I could come up so far is the dirac measure. In particular, I am looking for a measure $\mu$ ...
2
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1answer
23 views

Bernoulli product measure

Let $\Omega=\{0,1\}^\mathbb{N}$ and $\mathcal{A}$ the sigma-algebra generated by the cylinders sets $\{w\in\Omega\vert \forall s \in S, w_s=\epsilon_s\}$ with $S\subset\mathbb{N}$ finite and ...
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1answer
31 views

Doubling measure of an annulus

Recall that a doubling measure is a measure with the additional requirement that: $$\mu(B_{2R})\le C_\mu \mu(B_R)$$ for some contstant $C_\mu$. While solving some esercises related to doubling ...
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1answer
38 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
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1answer
180 views

Is $e^x$ finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $?

Does $e^x$ is finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $? I think it is, but I put on this question to make sure. I know $f$ being finite a.e. ...
2
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0answers
23 views

The dual space of weighted compact supported function?

Let $\Omega\subset \mathbb R^N$ be open bounded. It is well know that the dual space of $C_c(\Omega)$, i.e., compacted supported continuous function, can be identified by finite Radon measure ...
2
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45 views

Question on Egoroff-like theorem

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...
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2answers
128 views

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$?

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$? Think when we look at the equivalence classes of almost ...
2
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0answers
49 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
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2answers
3k views

Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$

I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
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1answer
47 views

How to use induction to show that $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n) $ are independent?

I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ ...
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5answers
127 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
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2answers
273 views

Probability space for stochastic processes

In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space ...
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2answers
19 views

Sigma algebra definition and Lebesgue integration

I know the definition of the $\sigma$-algebra, and I have seen it used in integration theory. However, I do not understand why it is defined the way it is. From what I understand, the definition ...
3
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2answers
37 views

Showing that a Borel Measure $\mu\equiv 0$

Problem. Let $\mu$ be a Borel measure on $[0,1]$. Assume that $\mu$ and Lebesgue measure $m$ are mutually singular. $\mu([0,t])$ depends continuously on $t$. $f\in L^{1}(\mu)$ for any ...