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

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3
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0answers
7 views

Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.

It is well known that a sphere minimizes the surface area to volume ratio since it reaches equality in the Isoperimetric Inequality. I'm trying to prove that no other closed surface has this property. ...
1
vote
1answer
13 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
0
votes
1answer
20 views

Measure Theory (Defining Measureability)

just a very basic measure theory question from the book by Bartle. Let (a) be the statement "For every $\alpha\in\mathbb{R}$, the set $A_\alpha=\{x\in X: f(x)>\alpha\}$ belongs to X." (X is a ...
4
votes
1answer
22 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
1
vote
1answer
24 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
2
votes
1answer
28 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
1
vote
1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
1
vote
2answers
32 views

Basic measure theory question about $\sigma$-algebra

Let $Y, Z$ be random variables and $G$ be a $\sigma$-algebra. Page 69 of Shreve's Stochastic Calculus for Finance II says "because both $Y$ & $Z$ are $G$-measurable, their difference $Y-Z$ is as ...
2
votes
2answers
19 views

A measure theory problem

Consider Lebesgue measure $\lambda$ restricted to the class $\mathscr B$ of Borel sets in $(0,1]$. For a fixed permutation $n_1, n_2, \dots$ of the positive integers, if $x$ has dyadic expansion ...
1
vote
1answer
15 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
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0answers
24 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
1
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1answer
35 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
3
votes
1answer
33 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
0answers
15 views

On extension of real valued functions from $(0,1)^{\infty}$ onto $[0,1]^{\infty}$

Let $\lambda$ be the standard Haar measure in $[0,1]^{\infty}$. Let $f$ be a bounded real-valued function on $(0,1)^{\infty}$ such that the set of all discontinuity points of $f$ has ...
2
votes
1answer
35 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
0
votes
0answers
23 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\emptyset \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K ...
1
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1answer
38 views

Which are the measurable sets with respect to this construction of an outer dirac-type measure?

Let $x \in \Omega$ and $\mathcal{A} \subset\mathcal{P}(X)$ an at most countable set with $\emptyset, \Omega \in \mathcal{A}$. Let $$ \delta_x: \mathcal{A} \to [0,\infty], \quad A \mapsto ...
4
votes
2answers
67 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
2
votes
2answers
19 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
0
votes
1answer
17 views

Probability, approximation by simple functions, boundedness and non-negativity.

In my probability class various results were announced requiring that a certain random variable was bounded. The specific example that interests me is the following: if $Y$ is $\mathcal{G}$-measurable ...
1
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0answers
19 views

Countable unions of Vitali sets…

Let $A \subset \mathbb{R}$ be sets of positive Lebesgue measure. Let $\Gamma$ be a countable dense subgroup of the additive group $\mathbb{R}$. Consider the partition of $\mathbb{R}$ canonically ...
3
votes
1answer
30 views

Transformation shift measurable

How to prove that this transformation is measurable? $\sigma:B(n)\rightarrow{B(n)} $ $\sigma(x)(k)=x(k+1)$ $\sigma(...,x_{-1},x_{0},x_{1},...)=(...,x_{0},x_{1},x_{2},...)$ where $B(n)$ with ...
1
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0answers
18 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
1
vote
1answer
20 views

Fatou's Lemma conditions for strict inequality

Under what conditions do we have equality (resp. strict inequality) in Fatou's Lemma? If the sequence $f_n$ is convergent, then it is obvious that equality holds. Is it the only case? There are some ...
-2
votes
1answer
61 views

Proposition on limsup

Suppose $\exists$ function $f: \mathbb{N} \to \mathbb{N}$ s.t. as $n \to \infty$, $f(n) \to \infty$. Prove that $\forall$ events (or sets) $A_1, A_2, ..., \limsup A_{f(n)} \subseteq \limsup A_n.$ ...
0
votes
0answers
31 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
votes
1answer
39 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 ...
2
votes
0answers
39 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
25 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 ...
0
votes
2answers
54 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 ...
3
votes
0answers
49 views
+100

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
26 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) = ...
0
votes
1answer
15 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 ...
1
vote
1answer
39 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 ...
2
votes
1answer
22 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 ...
0
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0answers
22 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 ...
0
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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 ...
1
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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 ...
1
vote
1answer
78 views
+50

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 ...
4
votes
0answers
35 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 ...
0
votes
2answers
19 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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votes
0answers
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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votes
0answers
23 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
39 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 ...
1
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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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votes
0answers
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
23 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 ...
2
votes
1answer
34 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.: ...
0
votes
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: $$\sum_{r=1}^nr{n\choose r}p^r(1-p)^{n-r}=\sum_{r=1}^nn{n-1\choose r-1}p^r(1-p)^{n-r}$$ ...