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

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Analogue of continuous mapping theorem

Suppose $X$ is a random variable defined on $[0,1]$ with probability density $f(x)$ for $x\in \mathbb{R}$. Based on a sample of size $n$, namely $X_1,\ldots,X_n,$ I defined an kernel estimator of ...
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4answers
81 views

Prove that if A is a non-(Lebesgue)measurable set and $d(A,B)>0$, show that $A⋃B$ is non-measurable

Prove that if A is a non-(Lebesgue)measurable set and $d(A,B)>0$, show that $A⋃B$ is non-measurable. $d(A,B)$ is the inf of distance $d(x,y)$ between two points $x\in A, y \in B$. I have tried ...
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3answers
39 views

Prove that if $A$ is a non-(Lebesgue)measurable set and $B⋂A=∅$, show that $A⋃B$ is non-measurable

The question is the following, Prove that if $A$ is a non-measurable set and $B⋂A=∅$, show that $A⋃B$ is non-measurable. If $B$ is measurable, then it is obvious since assuming $A\bigcup B$ ...
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2answers
37 views

Small derivative and the measure of a set.

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function, and that on some interval $(a,b)$, $|f'|\leq1$. Is it true that for all measurable sets $E\subset(a,b)$, ...
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3answers
105 views

What is the cardinality of the set of all non-measurable sets in $\Bbb R^n$?

The cardinality of the set of all measurable sets in $\Bbb{R}^n$ can be shown to be the same as the power set of $\Bbb{R}$ by looking into Cantor set. Denote $M=$$\{$$Ω⊆\Bbb{R}^n:Ω$ is ...
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2answers
40 views

What happens when we take a compliment in probability and why is sigma algebra needed?

When we take complement of a set, do we mean sigma algebra minus the set or only the sample space minus the set. Also why is sigma algebra needed in the axioms of probability ? For reference the ...
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Support of a discrete measure

According to Wikipedia (and some books), a discrete measure on the real line is a measure $\mu$ whose support $\text{supp}(\mu)$ is at most a countable set. This definition seems to be inconsistent ...
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1answer
19 views

A Change of Variable/Fubini's Theorem

A line in a text reads $$\int_{0}^{\infty} \mu (B(x,u^{\frac{-1}{t}}) du = t\int_{0}^{\infty} r^{-t-1} \mu (B(x,r)) dr.$$ I set $u=r^{-t}$. But then $du=-tr^{-t-1} dr$. Where is the negative?
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1answer
42 views

Proof that a random measure with orthogonal increments is a measure

Let me first state what I mean by a random measure with orthogonal increments. Definition: A random measure with orthogonal increments $Z$ is a collection $\left(Z(B): B \in ...
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1answer
127 views

A hard true or false question. “$M$ is measurable iff. for any subset $E\subseteq M$ we have $|M|_e=|E|_e+|M-E|_e$”

Can we say $M$ is Lebesgue measurable iff for any subset $E\subseteq M$ we have $|M|_e=|E|_e+|M-E|_e$? Here $|M|_e$ denotes outer measure. My feeling is that it cannot be right, or this is a very ...
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29 views

Condition for $\overline{M}$-measurable in problem 2.24 by Folland

I'm self-learning Real Analysis using Real Analysis of Folland, and I got stuck on this problem. Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X) < \infty$, and let $(X, ...
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2answers
52 views

Help hint on the following question regarding countable dense set and Lebesgue measure

Please help hint on the following question. Thank you! Let $E⊆\Bbb{R}^n$ be a measurable set with positive measure, and let $D⊆\Bbb{R}^n$ be a countable dense set. Prove that $|\Bbb{R}^n-⋃_{q∈D} ...
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2answers
63 views

How does the Vitali set violate the definition of measurable sets?

In my textbook the Vitali set is shown as a classic example of non-measurable sets. The proof is done by showing that you can derive an impossible measure of this set if it is measurable. I also ...
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0answers
22 views

Help with a proof regarding non-measurable set

Please help with the following question where $|E|$ denotes Lebesgue measure. We say set $A⊆\Bbb{R}^n$ is a translation of set $B⊆\Bbb{R}^n$ if $A=B+z$ for some $z∈\Bbb{R}^n$. Let $E$ be a ...
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0answers
91 views

I need to show a union is countable (most of proof is done)

Let: $\mathscr{J}^\circ_{\text{rat}}(\mathbb{R}^n)=\{(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)\subseteq\mathbb{R}^n|\ \forall i\in\{1,\cdots,n\}\ [a_i,b_i\in\mathbb{Q}]\}$ With the ...
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1answer
54 views

convergence of step functions in $L^1$ norm

Let $f \in L^1 (m)$. For $k=1,2,3,...$, let $f_k$ be the step function defined by $$ f_k (x) = k\int_{j/k}^{\frac{j+1}{k}} f(t)dt \ \text{ for $\frac{j}{k}<x<\frac{j+1}{k}$, ...
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2answers
62 views

measure theory problems and step functions

I have several questions that I haven't worked out. Any hints or solutions will be appreciated. Suppose that {$f_n$} is a sequence of real valued continuously differentiable functions on [$0,1$] ...
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1answer
59 views

Are all measure zero sets measurable?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} \ell(I_n): E \subset ...
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2answers
64 views

Is an $L^p$ function in an annulus $L^p$ restricted to almost all planes?

Let $n\geq3$ and consider the annulus-like domain $A=B(0,1)\setminus B(0,r)\subset\mathbb R^n$. Take any number $p\in[1,\infty]$. If $f\in L^p(A)$, is it true that $f|_{P\cap A}\in L^p(P\cap A)$ for ...
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1answer
21 views

Show the existence of decreasing set sequence $G_k$ st. $G_1⊇G_2⊇G_3⊇⋯⊇Ω$ and $|⋂G_k |_e=|Ω|_e$

The problem is Let $Ω$ be a set in $\Bbb{R}^n$. Show that there exists a decreasing sequence $G_k$ of open sets such that $G_1⊇G_2⊇G_3⊇⋯⊇Ω$ and $|⋂G_k |_e=|Ω|_e$. where $|Ω|_e$ denotes outer ...
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62 views

Understanding Rudin's proof that a Riemann integrable function is measurable

In the book "Principles of Mathematical Analysis" by Walter Rudin, he proves the following theorem (slightly reworded), Theorem. If $f$ is Riemann integrable on $[a,b],$ then $f$ is Lebesgue ...
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1answer
41 views

$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , where $X \sim N(0,1)$

Prove that: $$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}},$$ where $X \sim N(0,1)$ and $c>0$, $t \in\mathbb R$. The problem should be solved easily by using the equality: $$P(X \geq c) = P(e^{Xt} ...
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1answer
18 views

Complex Measures: Lebesgue

Given a Borel space $\Omega$. Consider a complex measure: $$\mu:\mathcal{B}(\Omega)\to\mathbb{C}$$ Regard a sequence: $$\eta_n\in\mathcal{L}(\Omega):\quad\eta_n\to\eta$$ Suppose one finds: ...
3
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1answer
31 views

Prove continuity of averaging function for integrable $f$

I want to prove the following statement which is part of a lemma in my textbook: Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le ...
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1answer
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Borel Measures: Pushforward

This thread is Q&A. Problem Given Borel spaces $X$ and $Y$. Consider a Borel measure: $$\mu:\mathcal{B}(X)\to\mathbb{C}:\quad\mu\geq0$$ Regard a pushforward: ...
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Difference between generator and the sigma algebra generated by this generator

Suppose $X$ is any set and $\mathcal{F} \subseteq 2^X $. By definition, I have learnt that $\sigma( \mathcal{F} ) $ is the smallest $\sigma$-algebra that contains $\mathcal{F} $. I am trying to ...
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1answer
51 views

Exchange of Limit and Integral with Nets

In topology, we have seen that there are examples of nets so that monotone and dominated convergence do not hold anymore. In particular, we worked with the net $\mathfrak{F}$ containing finite ...
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1answer
41 views

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar ...
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1answer
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Normal Operators: Von-Neumann

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}N\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Regard their algebra: ...
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Characterize the $\mu^*$- measurable sets where $\mu^∗ = \lambda^* \circ \text{proj}_1 $ and $ \lambda^*$ is the Lebesgue outer measure

Hi I'm working with Cohn's book and I have other problem with the necessity condition, I'd appreciate any help. Let $\lambda^*$ the Lebesgue outer measure on $\bf{R}$, and let $\pi$ be the ...
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1answer
31 views

Is the product $d x \otimes \mu_x(d y)$ on $[0,1] \times [0,1]$ a probability measure?

Let $Y=[0,1]$ and $X=[0,1]$ (both with the usual Borel sigma algebra). Let $\mu_x$ be a probability measure over $Y=[0,1]$ for each $x \in X$. (i.e $\mu_x$ depends on $x \in X$). Let $d x$ be the ...
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1answer
138 views

The dual of $\mathcal{L}^\infty$

Let $X$ be the measurable functions with finite supremum norm on the unit interval. This is a banach space with respect to the supremum norm. The continuous functions on the unit interval have as dual ...
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1answer
65 views

Haar measure on SO(n)

I am interested in describing the group of special orthogonal matrices SO(n) by a set of parameters, in any dimension. I would also like to obtain an expression of the density of the Haar measure in ...
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1answer
30 views

Doubling measure and Riesz Potential

I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows: Define the measure ...
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1answer
39 views

Proving a proposition regarding sigma algebras

Let $\{ X_{\alpha} \}_{\alpha \in A } $ be any collection and $X = \prod_{\alpha \in A} X_{\alpha} $. Let $\pi_{\alpha}: X \to X_{\alpha} $ be coordinate maps and let $\mathcal{M}_{\alpha} $ be sigma ...
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1answer
78 views

Integrable functions that take values in a Banach space

Let $\mathbb K$ be $\mathbb R$ or $\mathbb C$. Let $(X, \mathcal M, \mu)$ be a measure space and let $F$ be a Banach space over $\mathbb K$. I would like to define an integral of a function $f:X ...
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17 views

Borel Sets and Translations

Suppose $\mathcal{F}$ is Borel $\sigma$ algebra for a separable Banach space $X$ (i.e., potentially infinite dimensional). Is it obvious that for any $A \in \mathcal{F}$ and any $a\in X$, the ...
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31 views

On a step of a proof of the Borel-Cantelli lemma.

This is an excerpt taken form Probability with Martingales by Williams. The framework is probability theory. Why is the equation being discussed true if condition $\{ n \ge m \}$ is replaced by ...
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1answer
63 views

Subsets $B$ of bounded subinterval $I$ is lebesgue measurable iff $\lambda^*(I)=\lambda^*(B)+\lambda^*(I\cap B^c)$

Hi I was reading Cohn's book and I have problem with the following exercises (only the return of b is what I don't know), I'd appreciate any help and suggestion, if necessary, for a): a) Show ...
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Product measurable set induced by nonnegative function

Suppose $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f: X \rightarrow \mathbb{[0, \infty]}$ a nonnegative function. Let $G_f = \{(x,y) \in X \times [0, \infty]: y \leq f(x)\}$. ...
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1answer
42 views

Why do we need set $A$ to be countable in following proposition?

To define our terms: Let $\{ X_{\alpha} \}_{\alpha \in A } $ be any collection and $X = \prod_{\alpha \in A} X_{\alpha} $. Let $\pi_{\alpha}: X \to X_{\alpha} $ be coordinate maps and let ...
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1answer
71 views

Lebesgue measure of a subset of the unit circle

I'm having trouble getting started on this question: Let $S^1$ be the unit circle. Let $m = d \theta$ be the Lebesgue measure on $S^1$. Let $M \subset S^1$ be a measurable set such that $m(M) \geq 3 ...
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1answer
24 views

Using the dominated convergence theorem to bound the integral of a random variable

The following claim is used in the solution to problem 9.4 in Jacod and Protter's Probability Essentials: Claim: Let $X\in\mathcal{L}^{1}$ on $(\Omega,\mathcal{A},P)$ (where $\mathcal{A}$ is a ...
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1answer
41 views

The derivative of a measure

Let $\mu$, $\nu$ be two Radon Measure on $\mathbb{R}^n$. How can I prove that $D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))}$ is in $L^1_{loc}(\mathbb{R}^n,\mu)$?
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1answer
33 views

Showing a collection generates a sigma algebra

Let $X = X_1 \times X_2 $, let $\pi_i : X \to X_i $ ($i=1,2$) be coordinate maps. Let $\mathcal{M}_i $ be $\sigma-algebra $ on $X_i$. Let $\mathcal{F} = \{ \pi_i^{-1}(E_i) : E_i \in \mathcal{M}_i \} ...
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32 views

Definition of measurable functions whose values belong to a Banach space

Let $(X, \mathcal M, \mu)$ be a measure space and let $B$ be a Banach space over $\mathbb R$ or $\mathbb C$. It seems to me the most natural definition of a measurable function $f: X \rightarrow B$ is ...
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1answer
30 views

Existence of a Particular $L^1$ Function

I hope this question hasn't already been posted before in some other form (I couldn't find it, so if it has, please pardon me). I found this question on an old qualifier, but I am completely lost as ...
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2answers
67 views

porous sets: why measure zero?

We call a measurable set $E\subset\mathbb R^N$ porous if every ball $B_r(x)$ contains a smaller ball $B_{cr}(y)$ for some $c\in(0,1)$ such that $$ B_{cr}(y)\subset B_r(x)\setminus E. $$ So I've read ...
3
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3answers
71 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
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1answer
37 views

Is every continuous CDF the limiting distribution of some sequence of discrete CDFs?

Note: I know that for various measure-theoretic reasons (that I don't fully understand) this does NOT apply to the underlying probability density. I'll accept as answers either a proof, paper to a ...