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

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8 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 ...
2
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3answers
536 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
1
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1answer
20 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 ...
0
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0answers
12 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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1answer
52 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 ...
1
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0answers
12 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 ...
0
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2answers
52 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 ...
0
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1answer
29 views

Proposition on limsup

Given sets (or 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 ...
1
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1answer
19 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 ...
1
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0answers
12 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 ...
6
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1answer
1k views

Proving the measure of an increasing sequence of measurable sets is the limit of the measures

Show that if $A_1\subseteq A_2\subseteq A_3\cdots$ is an increasing sequence of measurable sets(so $A_j\subseteq A_{j+1}$ for every positive integer $j$),then we have ...
2
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1answer
32 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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1answer
38 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
58 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
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0answers
27 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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0answers
33 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{ ...
6
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1answer
378 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
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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
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1answer
122 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this? I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb ...
2
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1answer
24 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
25 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) = ...
30
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2answers
547 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
0
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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 ...
2
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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 ...
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 ...
0
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0answers
18 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

Fubini's theorem on a product of locally compact spaces which do not have countable bases

Let $X$ be a locally compact Hausdorff space. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open subsets of $X$. A measure $\mu$ on $\mathcal B$ is called a (positive) Radon ...
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2answers
192 views

Determining a measure through a class of measure preserving functions

Let $\mu$ and $\mu^\prime$ be probability measures over the sigma algebra $\Sigma$ consisting of the Lebesgue measurable subsets of $[0,1]$. Suppose also that $\mu$ and $\mu^\prime$ assign measure $0$ ...
4
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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 ...
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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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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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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.
4
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2answers
1k views

Meaning of convergence in $L^1$ norm

What does it mean for a sequence $f_n$ to converge to some function, say, $f$ in the $L^1$ norm? Is it enough to show that $\int|f_n -f| \to 0$ or must one show as well that $f\in L^1$? I am ...
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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 ...
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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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 ...
2
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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
20 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 ...
1
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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 ...
0
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1answer
50 views

$\lim_{y \rightarrow^{nt}x}\int \omega(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$

We need to show $\lim_{y \rightarrow^{nt}x}\int \omega_j(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$ Here is the necessary information ...
0
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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: $$\sum_{r=1}^nr{n\choose r}p^r(1-p)^{n-r}=\sum_{r=1}^nn{n-1\choose r-1}p^r(1-p)^{n-r}$$ ...
2
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1answer
919 views

lower bound for probability distribution of a random variable

If $X$ is a random variable with finite mean $\mu$ and variance $\sigma^2$, how do I show that the estimate \begin{equation*} P[\mu − d\sigma < X < \mu + d\sigma] ≥ 1 − 1/d^2~\forall d>1 ...
0
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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 ...
0
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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 ...
0
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1answer
48 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
1
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1answer
50 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 ...
1
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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 ...
4
votes
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 $\int_Kx \,dx \geq \int^{m(K)}_0 x\,dx$, ...