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

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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 ...
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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 ...
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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 ...
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0answers
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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 ...
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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 $c < ...
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Help explain why “$f$ be finite a.e” is necessary in a theorem.

A theorem states Let $ϕ$ be continuous on $\Bbb{R}$, let $f$ be finite on $Ω$ a.e., then $ϕ∘f$ is measurable if $f$ is measurable. The following is the proof from my textbook. Since $ϕ$ is ...
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4answers
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Can anyone give an example of a closed that contains no interval but with finite non-zero Lebesgue measure?

Can anyone give an example of a closed set $F$ of $\Bbb{R}$ such that $0<|F|<+\infty$ and $F$ contains no open interval? Thank you!
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1answer
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Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
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2answers
29 views

Restriction of measure arising in Riesz's theorem to Borel sets

Riesz's theorem on representation of positive linear functional on locally compact space as stated in Rudin's "Real and Complex Analysis" assures us that certain $\sigma-$algebra containing all Borel ...
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1answer
28 views

Determine if $\left| {\mathop {\lim \sup }\limits_{n \to \infty } {A_n}\Delta \mathop {\lim \sup }\limits_{n \to \infty } {B_n}} \right| = 0$

The question is Is $\left| {\mathop {\lim \sup }\limits_{n \to \infty } {A_n}\Delta \mathop {\lim \sup }\limits_{n \to \infty } {B_n}} \right| = 0$ if given $\left| {{A_n}\Delta {B_n}} \right| = ...
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Brownian motion needs to be defined continuous for every $\omega$ to be jointly measurable.

Let $B=(B_t)_{t\in[0,\infty)}$ a Brownian motion (BM) and $(\Omega,\mathcal{F},P)$ be the probability on which $B$ is defined. Some define BM as a.s. continuous, e.g., Brownian motion is almost surely ...
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If $f$ and $g$ are measurable and well-defined a.e., show $f+g$ is measurable. [duplicate]

The question is If $f$ and $g$ are measurable and well-defined a.e., show $f+g$ is measurable. I can show that the case when $f+g$ are well-defined everywhere. Let $\Omega_{f+g>a}=\{x\in ...
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1answer
69 views

Continuous and measurable in each variable $\implies$ product measurable?

Consider a metric space $A$ with a metric $d$, and consider the measurable space $(A,\mathcal{B}(A))$ with the Borel $\sigma$-algebra generated by $d$-open sets. Let $(\Omega,\mathcal{F})$ be a ...
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2answers
23 views

If $M$ is $F$-measurable, then is it also $F'$-measurable with $F'\subset F$?

$F$ and $F'$ are $\sigma$-algebras, and $M$ is a function from $(\Omega,F)$ to $(\mathbb{R},B(\mathbb{R}))$ If this statement is true, how to reason or understand it in a simple way?
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55 views

Equivalent conditions of Lebesgue measurable sets

Hi I'd appreciate if someone can check the following exercise any suggestions are welcome. Thanks ;) Let $A$ a subset of ${\bf{R}}^d$ show that the following conditions are equivalent: (i) $A$ ...
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1answer
26 views

Relation between Lebesgue measure of two sets

question in lebesgue measure: Given that $T$ is a Jordan set of positive Lebesgue measure, $l(T)>0$. If $M \subset T $ such that $l(M)=0$ where $l(\cdot)$ denote Lebesgue measure, is it true ...
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Generalized convex combination over a Banach space

The Question: Is the following true? If not, what further hypotheses do I need? Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, ...
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1answer
29 views

$L^p$ and $\ell^p$ spaces

I'm confused. I've read that for $1\leq p<q<\infty$ following inclusions are true: $$\mbox{1)}\qquad \ell^p\subset\ell^q$$ $$\mbox{2)}\qquad L^q\subset L^p$$ My question is - why inclusions ...
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Construction of a set with density of half at $0$.

If we define for a given set $A \subset \Bbb{R}$ and $x\in \Bbb{R}$ the density of $A$ at $x$ being the limit as $[I]$ goes to zero of the ratio $[I \cap A]/[I]$ wherever the limit exists for ...
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1answer
69 views

Is the following set closed?

Let $\Omega$ be a convex and bounded set with a finite diameter, $\text{diam(}\Omega) < \infty$, and let $f$ be in $L^1(\Omega)$ which is positive almost everywhere. Then, the Lemma (from the ...
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0answers
20 views

Integral as countably additive map

Let $(\Omega, \mathcal{A}, P)$ be a probability space. Then is $\mu : \mathcal{A}\to \mathbb{R}$ defined by $$\mu(A)=\int_{A}X\,\mathrm{d}P$$ a countably additive set function, i.e. for $A_j\subseteq ...
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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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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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38 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
107 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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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
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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
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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 $\left(Z(B): B \in ...
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1answer
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A hard true or false question. “$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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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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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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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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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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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
55 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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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
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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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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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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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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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$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , 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: ...
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1answer
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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
16 views

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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2answers
28 views

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 ...
5
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1answer
52 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
40 views

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: ...