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

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Measurability of Dini Derivatives

Let $f:(0,1)\to\mathbb R$ be measurable. Then, the (right upper) Dini derivative $$ D^+ f(x) = \limsup_{h\to 0^+} \frac{f(x+h) - f(x)}{h} $$ is also measurable (a well known result of Banach). Can ...
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1answer
33 views

How to write down the probability space of this stochastic process

Consider infinitely repeated coin-toss. Then the probability space can be written as $\Omega=\{H,T\}^\infty$ with its product $\sigma$-algebra. Now let's assume that after each round, there is ...
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2answers
131 views

Prove that jump functions are measurable

This question comes from the exercises of Stein and Shakarchi's Real analysis Ex. 5.14. Define $$ j_n(x)= \begin{cases} 0& \text{if } x< x_n\\ \theta_n & ...
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0answers
21 views

Dirac Measure is Purely Atomic

In my book, "Probability and Stochastics" by Cinlar, it's stated that for some measurable space $(E,\scr E)$, and fixed $x\in E$, the Dirac measure $\delta_x(A)=\left\{ \begin{array}{lcc} ...
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3answers
65 views

Lebesgue outer measure of disjoint sets in $\mathbb{R}^n$

If $d(A, B) > 0$, then it's true that $m^*(A\cup B) = m^*(A) + m^*(B)$. If there are disjoint open sets $U, V$ such that $A \subset U$ and $B \subset V$, doesn't it still hold that $m^*(A\cup B) = ...
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2answers
101 views

Exercise of measure theory

I need help with this exercise. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $(A_n ;n\geq 1)\subseteq \mathcal{F}$, such that $\mu (\bigcup_{n=1}^{\infty}A_n)<\infty$ and ...
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1answer
28 views

Skorokhod space with uniform norm is Banach

Let $D := D([0,t])$ be the Skorokhod space of right-continuous functions with left limits taking values in $\mathbb{R}^d$. Equip $D$ with the supremum norm $||f||_\infty = \sup_{s \in [0,t]}|f(s)|$. ...
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1answer
52 views

Why is the support of Dirac distribution $\{0\}$?

Distributions are of two types: those that are obtained from locally integrable functions, and those that aren't. For the first type, the support of distribution is simply the support of the function. ...
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1answer
67 views

Prove that this function is not Riemann-integrable

Let $A\subset[0,1]$ be the union of open intervals $(a_i,b_i)$ such that each rational number in $(0,1)$ is contained in some $(a_i,b_i)$ and $\sum_i(b_i-a_i)<1$. It can be shown that $\partial ...
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0answers
59 views

Strong law of large numbers

Suppose $X_i\in\mathcal{L}^2$ with expectation $0$ such that $\sum_{i=1}^\infty \mathbb{E}[X_i^2]/i^2<\infty$ and suppose they are pairwise non correlated. Does then the SLLN still hold?
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81 views

$f(x) > 0$ , prove that $\int{f} > 0$

Let $f$ be a function such that $ \forall{x}, f(x) > 0 $ and is integrable in $[a,b]$ prove that: $ \int_a^b{f(x)} > 0$ or show a counter example
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20 views

Properties of the Kernel from the measurable space $(X,\mathscr{A})$ to $(Y,\mathscr{B})$

Hi everyone this is an exercise from Cohn's book. I'd appreciate if someone can check part (d) and (e) where I have more problems because this concept is completely new for me. Let $(X,\mathscr{A})$ ...
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1answer
129 views

New definition of Lebesgue integral

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
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1answer
43 views

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y)dy = 0$ for all $x$. Then $f = 0$ a.e.

This is a corollary to a proof in Bass, but I don't understand why it follows from the proof he gives. I follow everything up until the last statement. Why is it that proving that the integral is $0$ ...
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0answers
15 views

Doubt regarding limitting value of partial derivative of $C^{1}$ function

The aim is to prove the following result: Let $v : \mathbb R \to \mathbb R$ be such that: $v \in C^{1}(\mathbb R)$ & $|\frac{\partial v}{\partial x}| \in L^{1}(\mathbb R)$ . Then to prove that: ...
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1answer
36 views

If a set of functions contains a sequence that is Cauchy, but not convergent, what does that imply about the set or the functions?

This is not homework. I am studying a set $S$ of of distributions that all have a fixed mean and also satisfy some ancillary criteria. Lets say I have a sequence of probability density functions ...
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1answer
11 views

Interpretation of proof of the Lebesgue-Stieltjes measure as a $\sup$ over compact subsets

The following is part of a theorem and proof in Folland's Real Analysis: Modern Techniques and Their Applications: Let $\mu$ be a complete Lebesgue-Stieltjes measure on $\mathbb{R}$ associated to the ...
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1answer
79 views

Under what conditions does a specified conditional distribution exist

It is common to see conditional distributions specified in stats like: $$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$ (Of course, we can also use some other distribution here) How do you prove that ...
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1answer
21 views

Calculate the measure of a measurable set under nonlinear mapping.

It is known that: If $\cal{A} \subset \mathbb{R}^n$ is Lebesgue measurable, and $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear mapping, then $L(\cal{A})$ is Lebesgue measurable and ...
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1answer
26 views

Algebraically solve a component in a partial sum

I would like to solve this equation for y: $$T = -a + \sum_{1}^{n} \frac{\left(\frac{x}{n} - \frac{y}{n} \right)}{ (1+b)^{n} }$$ The partial sum (Σ) is from 1 to n. I use the ^ symbol for an ...
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1answer
39 views

Lebesgue measure of a set of real numbers well-approximated by rationals

Consider the set $A$ of real numbers $r$ such that there exists a constant $C$ and sequence $\frac{p_n}{q_n}$ of rational numbers (where $p_n$ and $q_n$ are integers) with $q_n \rightarrow \infty$ and ...
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0answers
25 views

Integral of sup of directed family of elementary functions

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
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1answer
35 views

Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
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1answer
35 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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1answer
67 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 $Y$ with respect to the uniform spherical ...
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0answers
23 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
63 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
43 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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2answers
70 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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2answers
137 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
25 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 . ...
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1answer
92 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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1answer
57 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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1answer
77 views

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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157 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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31 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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50 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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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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1answer
26 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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0answers
28 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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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
16 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, ...
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1answer
26 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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0answers
24 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 ...
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2answers
85 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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61 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 ...
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1answer
36 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 ...
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1answer
27 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
37 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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2answers
48 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 ...