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

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Proposition on limsup

Given events $A_1, A_2, A_3, ...$ and function $f: \mathbb{N} \to \mathbb{N}$, show that $\limsup A_{f(n)} \subseteq \limsup A_n \Leftrightarrow f(n) \to \infty$ This is what @Did told me last year. ...
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22 views

A Measure Problem on Stein's Real Analysis

I'm considering problem 5 on Stein's real analysis chapter 6 $X$ is a metric space, for any positive linear functional $l$ on $C_0 (X)$ which are the continuous functional on $X$ supported in some ...
2
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1answer
24 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
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0answers
30 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
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0answers
21 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
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2answers
39 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
2
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1answer
17 views

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
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0answers
22 views

Measurable maps in metric spaces.

i have several questions about measurability of maps with values in metric spaces : 1/ When $X$ and $Y$ are two separable metric spaces, it is easy to prove that $\mathcal{B}(X\times Y) = ...
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1answer
14 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
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1answer
35 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
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1answer
19 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
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17 views

Conditionals of signed measures

My question pertains the definition of regular conditional measures of signed measures defined on product spaces. Consider a Suslin measurable space $\mathcal A=X\times Y$ with the Borel ...
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0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
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0answers
21 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...
2
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1answer
35 views

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
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34 views

$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
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2answers
17 views

Expected Values of the product of a random variable and an indicator random variable

Let $X$ be a random variable $\in$ $L_{1}$ Given that $E[X]$ = $1$ , does that necessarily mean that : $E[X*1_{A}]$ = $P[A]$ ? My intuition is yes, since this is can be decomposed to $E[X]$ * ...
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17 views

Estimate for measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function (https://en.wikipedia.org/wiki/Nevanlinna_function) and let $\mu$ be the measure associated to $F$ via the Stieltjes inversion formula: ...
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22 views

definition & properties of Lebesgue function [on hold]

I need the correct definition & properties of Lebesgue function to answer one question using Lebesgue function.
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1answer
36 views

Three questions about Haar measure

I have been reading on Haar measure recently. Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ then $G$ is discrete. $\mu(G)<\infty$ then $G$ is compact. we know ...
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1answer
17 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
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21 views

Continuous function such that range is different from the essential range.

Let $f$ be a function $\mathbb R \to \mathbb R$. The essential range EssRan(f) of $f$ is defined as the set of all numbers $z$ such that the preimage of every open ball around $z$ under $f$ has ...
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2answers
18 views

Are as constant but not constant random variables trivial sigma-algebra-measurable? Converse?

Are almost surely constant random variables trivial sigma-algebra-measurable? These links suggest no: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2004&task=show_msg&msg=1121.0001 ...
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0answers
35 views

$\sigma$-algebra

I have a question about $\sigma$-algebra. Let $(S,\Sigma)$ be a measurable space. Let $A \in \Sigma$. We can define $A \cap \Sigma:=\{A \cap M:M \in \Sigma\}$ and $A \cap \Sigma$ is $\sigma$-algebra ...
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1answer
33 views

A piecewise $C^1$ curve has Jordan measure zero.

$\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: ...
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2answers
30 views

Proving a sequence of numbers in binomial

Consider the set $P_r={n\choose r}p^r(1-p)^{n-r}$ Prove that: $$\sum_{r=1}^nrP_r=np$$ By far I attempted: $$\sum_{r=1}^nr{n\choose r}p^r(1-p)^{n-r}=\sum_{r=1}^nn{n-1\choose r-1}p^r(1-p)^{n-r}$$ ...
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0answers
24 views

Orthonormal List In Hilbert Space

guys say we have a orthonormal list say ${O_n}_{n\in A}$ that is an orthonormal list in a Hilbert Space $X$. Is it true that the list is complete if and only if $<a,b>= \sum_{n \in A} ...
4
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1answer
31 views

Finding a Radom-Nikodym derivative

Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1,f_2\in L^1(\mu)$ and consider the signed measures $$v_i(E):=\int_Ef_id\mu$$ for every $E\in\Sigma$. If $v_1\ll v_2$ and $v_2\ll v_1$, we must find ...
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1answer
8 views

Question about the positive variation of a signed measure

If $(X,\Sigma)$ is a measurable space and $v$ is a signed measure, I want to prove that $$v^+(E)=\sup\{v(F):F\subseteq E,F\in\Sigma\}$$ where $E\in\Sigma$. Let $X=P\cup N$ a Hanh descomposition for ...
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1answer
45 views

Is every element contained in a smallest measurable set?

Let $(X,\mathcal F)$ be a measure space, then for each $x \in X$ does there always exists a smallest measurable set containing $x$? If $X$ is countable or $\mathcal F$ finite, then this is true, as ...
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1answer
49 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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0answers
28 views

Continuity of quantiles as function of measure.

Let $P(R)$ be the probability measures on the real numbers $R$ and fix $\alpha \in (0,1)$. Define $$Q_{\alpha} : P(R) \to R $$ as the function taking a measure $\mu \in P(R)$ to its $\alpha$-th ...
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0answers
20 views

Haar Measure on Locally Compact monoids

I have been reading on Haar measure and we know that every locally compact Hausdorff group admits a Haar measure, is the same true for semigroups with identity $e$(monoid)? If not, is there a class of ...
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2answers
23 views

Derivation of the Hypergeometric Distribution

The derivation of the hypergeometric refers to the following example: An urn contains N white balls, M black balls and we draw $n\le N+M$ balls without replacement. Let X be the number of white balls, ...
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1answer
24 views

Measure Theoretic Definition of a Random Variable

I am struggling a little with the definition of a RV: Let $(\Omega,F),(\Omega',F')$ be two event spaces. Then every mapping: $X:\Omega \to\Omega'$ is a RV provided $X^{-1}A' \in F,~~~ \forall A' \in ...
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47 views

An application of the uniform boundedness principle

Can someone provide me with a sketch of proof / hint for this exercise: Let K $\subset$ $L^1([0,1]\,,\, \mu)$ be a closed linear subspace. If $\forall$ $f \in K$, $\exists \, p > 1$ such that $f ...
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2answers
27 views

Injective implies invertible? Injective and well-defined implies bijective?

I have two questions regarding functions regarding linear maps: (Let $X$ and $Y$ be to Banach spaces) If $T:X\rightarrow Y$ is injective, then $T^{-1}$ exists, right? If $T:X\rightarrow Y$ is ...
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1answer
21 views

Lebesgue outer measure with open balls

I'm not sure if this has been asked before; if so please redirect me to the appropriate question. The Lebesgue outer measure of $A \subseteq \mathbb{R}^n$ is defined as $$\mu_*(A) = \inf\left\{ ...
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2answers
28 views

Infinite sigma-algebra is uncountable

The problem is so well-known, but I still do not understand some points in the following solution: Why is every member $S$ of $\mathcal{S}$ a union of the sets $f(x)$ with $x\in S$ ? What can we ...
4
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1answer
31 views

integral over a subset of interval in $\mathbb{R}$

Consider a finite interval $[0,d]$, where $d$ is a positive real number. Let $K$ be a measurable subset of $[0,d]$ Then, how can I prove or disprove that $\int_Kx \,dx \geq \int^{m(K)}_0 x\,dx$, ...
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Show that if $P(Y_1 \neq 0) > 0$, then with probability one, $\limsup\limits_{n} X_n = +\infty$, $\hspace{10mm}\liminf\limits_{n} X_n = - \infty$ [on hold]

I'm having quite a bit of difficulty with the following problem. Any answer or detailed explanation would be greatly appreciated. Let $Y_1, Y_2, ...$ be bounded iid random variables such that $EY_1 ...
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1answer
18 views

Non measurable subset of a positive measure set

I am self-studying measure theory and I have seen this theorem: If $A$ is a set of positive measure, then there exists a subset $D$ of $A$ that is non measurable. I am not sure how to prove it. I ...
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1answer
15 views

Basic doubt about measurable subspace

I was having some doubts related to measure subspace of a given space $(X,\Sigma,\mu)$. I've read that if one defines $\Sigma_E=\{A \cap E: A \in \Sigma\}$, then $\Sigma_E$ is a $\sigma-$algebra of ...
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1answer
16 views

Is there an example of a bounded connected set that is not Jordan measurable?

Like the title asks, is there an example of a set that is connected and bounded that is not Jordan measurable?
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16 views

Multiplicative structure of a family of functions and $\sigma$-algebra

I have a question about a multiplicative structure and $\sigma$-algebra. Let $S$ be a set and $\mathcal{J}$ be a familly of $\mathbb{R}_{+}(:=[0,\infty[)$-valued bounded functions on $S$, having the ...
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1answer
31 views

Proving basic properties of Hausdorff dimension and measure

I have two questions on basic properties of the Hausdorff measure and dimension which I've taken for granted for a while (I'm revisiting Falconer after about a year), but that I've never actually seen ...
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1answer
19 views

Examples of predictable processes

I am asked to prove that the following processes are predictable. I am used to looking at stochastic processes as sequences of random variables (by fixing time) or as a collection of paths (by fixing ...
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13 views

Completion of a stochastic basis (Filtration)

Given a stochastic basis ($\Omega, \mathbb{F},(\mathbb{F_t})_{t \in \mathbb{R}},P$) with a right-continuous filtration, it is possible to construct a complete stochastic basis $\Omega, ...
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1answer
34 views

Show a subset of $[0,1]^2$ has content $0$

I've been studying Spivak's "Calculus on Manifolds" and I'm thinking about the following question: Let $C \subset [0,1]^2$ be the union of all $\{p/q\} \times [0,1/q]$, where $p/q$ is a rational ...
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1answer
61 views

Trouble understanding what a measure-zero set is.

To begin with some context, I haven't had any exposure to measure theory yet. I solved the following problem. A set $A\subset \mathbb R$ such that $\forall \epsilon >0$, there exists countably ...