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

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153 views

Measure & integration vs Complex analysis

If I want to go down a statistics (masters degree) track that's a bit heavy on the math side, and I had to choose between complex analysis and measure theory as a course which one should I take and ...
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97 views

A particular measure in the Cantor space $2^\infty$ / How to prove it also defines a $\sigma$-algebra?

Consider the following measure $\mu$ for the Cantor set (seen as the space of infinite sequences of 0's and 1's): $$ \mu\left(E\right) = \lambda \left(g\left(E\right) \right) \tag{1}$$ where ...
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33 views

Integrals agree over different sets implies integrands agree.

I am in a situation where $f$ is $\Sigma$-measurable, and for every $A \in \Sigma$ I know that $\int_A f = \int_A g$. I know $f,g \in L^1$ are measurable with respect to the larger $\sigma$-algebra ...
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45 views

$m^*(A) = m^*(A + t)$

Define $m^*(A) = \inf Z_A$ as the outer measure of $A \subseteq \mathbb{R}$ where $$Z_A = \left\{\sum_{n=1}^{\infty}|I_n| : I_n \text{ are intervals}, A \subseteq \bigcup_{n=1}^{\infty}I_n\right\} ...
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349 views

Continuity of the Lebesgue function

If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ ...
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161 views

What almost sure convergence means in the context of strong law of large numbers

According to http://en.wikipedia.org/wiki/Almost_sure_convergence#Almost_sure_convergence, a sequence of random variables $X_n$, which are a function of a shared sample space $Ω$, is said to converge ...
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74 views

Atoms in a countable space

Let $(\Omega, \mathcal{F})$ be a measurable space where $\Omega$ is countable. I am trying to prove that there is some partition $\mathcal{P}$ of $\Omega$ such that the $\sigma$-algebra ...
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88 views

Uniqueness of a measure extension as a corollary

Let $\mu$ be a $\sigma$-finite premeasure on a semiring $S$ and $\nu$ the outer measure generated by it. Supposing I know that for each extension $\nu'$ of $\nu\big|_{\sigma(S)}$, that has the ...
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531 views

Difference between $\prod $ and $\bigotimes$

What is the exact difference between $\prod $ and $\bigotimes$ when both are taken over collection of sets? e.g. $\{A_i\}_{i\in I}$ be a collection of sets. What are $$\prod_iA_i \text{ and ...
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482 views

Convergence implies lim sup = lim inf

Could someone please explain to me how the following can be proven? I get the intution but don't know how to write it rigorously. Thank you.
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98 views

What is the algebra generated by the class $\mathcal{E}$ of sets described?

What is the algebra generated by the class $\varepsilon$ of sets described? a) For a fixed subset $E$ of $S$, $\mathcal{E}=\{E\}$ is the class containing $E$ only. b) For a fixed subset $E$ of $S$, ...
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3answers
113 views

Upper bound on the integral of a function given the measure of the integration domain

Let $(X,\mathcal{M},\mu)$ be a measure space and $f:X\rightarrow\mathbb{C}$ be a measurable function such that $\int_X |f|\; d\mu <+\infty$. I am trying to prove the following result: ...
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2answers
130 views

$E^p$ is a ring, but not a $\sigma$-ring

An interval in $\mathbb{R}^p$ is defined as a set of the form $\{ \vec{x} \in \mathbb{R}^p : \forall i=1,2,...,p \quad a_i \le x_i \le b_i \}$, where the inequalities can be strict or not, and the ...
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56 views

Show: $\mathcal{G}:=\left\{B\in\mathcal{B}(\mathbb{R}^n)|t+B\in\mathcal{B}(\mathbb{R}^n)\right\}$ is $\sigma$-Algebra

Let $t\in\mathbb{R}^n$. Show that $$ \mathcal{G}:=\left\{B\in\mathcal{B}(\mathbb{R}^n)|t+B\in\mathcal{B}(\mathbb{R}^n)\right\} $$ is a $\sigma$-Algebra. I have to show: 1) ...
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77 views

Clarifying a step in proving uniqueness of Jordan Decomp of signed measures

This should be real simple but I have been struggling to see this in an easily intuitive manner. Basically, my confusion comes down to showing that if $A,B$ and $A',B'$ are two arbitrary Hahn ...
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63 views

Convergence in metric and in Borel measure

Suppose we have the space ($\mathbb{R}, B(\mathbb{R},\lambda$) and define e $h: \mathbb{R} \rightarrow \mathbb{R}$ by $h(t) = 1/(1+t^2)$. a) Prove that the formula $$d(f,g) = \int ...
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1answer
50 views

Does the squared root $\sqrt{|\cdot|}$ belong to $BV((-1,1))$?

Does the squared root belong $\sqrt{|\cdot|}$ to $BV((-1,1))$? In an affirmative case, what is its derivative in the distributional sense, i.e., what is the Radon measure $\mu$ such that $\mu=Du$ ?
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71 views

Does $f \in L(E) \rightarrow$ that $f$ is measurable?

So this arose because a proof I was going over stated that since $f\in L(E)$ then $f$ is measurable. This wasn't immediately clear to me. I guess that Lebesgue integration, at least in Royden, is ...
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122 views

Weak-* convergence of Borel probability measures on a nice space

Let $X$ be compact, metrizable space. (If it somehow changes the outcome of the question, I am mainly interested in $X=$ the Cantor set) Let $E\subset X$ be a closed subset and let $\mu_n$ be a ...
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37 views

Partitions of a simple function, doesn't make sense.

Let $(\Omega, \mathcal{F})$ be a measurable space and let $A_i,B_j$ be in $\mathcal{F}$ such that $\Omega = \uplus_{i=1}^n A_i = \uplus_{j=1}^m B_j$, ie they're finite disjoint unions. Let $s$ be a ...
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61 views

Defining a measure $\lambda$ in terms of a sequence of measures $(\mu_n)$

Below is a problem concerning a sequence of measures. My trouble arises when trying to prove the countable additivity of our measure $\lambda$, which is defined in terms of a sequence of measures. (I ...
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36 views

Change of variables with a square

Can someone help me understand this a bit better: $\int (x-y)^2 dx = \int(y-x)^2dx$ as $(y-x)^2 = (x-y)^2$. Now, if I make the change $z = x-y$ in the one on the LHS I get: $\int z^2 dz$ as $dz ...
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57 views

Measurability of section functions

Let $(X,\mathcal{A})$ be a measure space and $K$ a compact metric space wiht the Borel $\sigma$-álgebra $\mathcal{B}$. Is it true that if $f:X\times K\rightarrow \mathbb{R}$ is ...
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49 views

Measurable subset of a product space has measurable sections

Let $(F, \mathcal{F})$ and $(G, \mathcal{G})$ be measurable spaces. How can we show that if $E$ is measurable with respect to $\mathcal{F}\otimes \mathcal{G}$, then for every $x\in F$, the set $\{y\in ...
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83 views

Quick question about $\sigma$-algebras.

I have a quick question concerning $\sigma$-algebras. If A is a collection of subsets of a set X and Y is the $\sigma$-algebra generated by A, then can I conclude that every element of Y is either (1) ...
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106 views

Integral of sum of sequence of integrable functions

Let $f_n$ be integrable function and $f_{n}^{+},f_{n}^{-}$ be its positive and negative parts. Are these steps correct? \begin{align} \int {\sum {f_n}} &=\int{\sum{(f_{n}^{+}-f_{n}^{-})}}\\ ...
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35 views

The difference of two charges

Definition in Bartle: If $\mathcal{X}$ is a $\sigma$-algebra of subsets of a set $X$, then a real-valued function $\lambda$ defined on $\mathcal{X}$ is said to be a charge in case $\lambda(\phi)=0$ ...
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306 views

“Probability” measures on Cantor set

I'm not fully acquainted with measure theory, so a detailed explanation may be needed here. From what I already understand, the Lebesgue measure on Cantor set (denote it by: CL-measure) $C$ gives ...
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201 views

Double expected value

Let $m$ be a probability measure on $\mathbb{R}^n$, so that $m(\mathbb{R}^n) = 1$. Consider two measurable functions $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and $g : ...
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89 views

Convergence almost surely

Let $X_n$ and $X$ be random variables. If $X_n \to X$ almost surely, then we have that $$ \mathbb{P}\left( \lim_{n \to \infty} X_n = X\right) = 1. $$ My question is, can we conclude that $$ \lim_{n ...
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34 views

$F(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)$ is integrable

I'm studying for an exam and I've encountered this exercise: Let $f(x)=x^{-0.5}\mathbb{1}_{\{0<x<1\}}$ and ${\{r_n\}}_{n=1}^{\infty}$ some enumeration of the rationals. Let: ...
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336 views

metric and measure on the projective space

Let $RP^n$ be the $n$-dim real projective space. I have the following four questions. What is the so called standard metric on $RP^n$? More generally, consider a metric space $M$ with an equivalent ...
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171 views

Convergence of increasing measurable functions in measure?

Let ${f_{n}}$ a increasing sequence of measurable functions such that $f_{n} \rightarrow f$ in measure. Show that $f_{n}\uparrow f$ almost everywhere My attempt The sequence ${f_{n}}$ converges to ...
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69 views

equivalent measures, can be one finite and one not?

Let $\mu$ be a non-negative and Borel-finite measure on $\mathbb{R}$ and $\nu$ a non-negative measure on $\mathbb{R}$. If $\mu$ and $\nu$ are equivalent (one absolutely continuous with respect to the ...
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4answers
413 views

$m(\alpha E)=\alpha m(E)$ ? for every lebesgue measurable set and $\alpha >0$

Let's consider a lebesgue measurable set $ E \subset \mathbb R$. And let's consider a positive constant $\alpha>0$. I want to know if it's always true $m(\alpha E)=\alpha m(E)$. Clearly $m$ denote ...
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98 views

Showing measurability of a function (integrals and Bochner space)

Suppose $V$ is Hilbert space and I know that for every $u \in L^2(0,T;V)$, the integral $$\int_0^T \langle f(t),u(t)\rangle_{V^*,V}$$ exists (because it equals another integral, and I know that ...
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152 views

exercise 28 chapter 1 in stein shakarchi

I do exercise 28 in chapter 1 of Stein and Shakarchi and they say that for any $E \subseteq \Bbb{R}^d$, $0 < \alpha <1$ we can find an open set $O \supseteq E$ such that $m_\ast(E) > \alpha ...
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67 views

Ham sandwich for measuers implies the classical one

Ham sandwich theorem for measures: Let $\mu_1,\mu_2,\mu_3 $ be finite Borel measures on $\mathbb{R^3}$ such that every hyperplane has measure $0$ for each of $\mu_i$. Then there exists a ...
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515 views

Proof of the Lebesgue-Radon-Nikodym Theorem

Theorem: Let $\lambda, \mu$ be $\sigma$-finite measures defined on the $\sigma$-algebra $\mathcal{A}$ of the space $X$. Then, a) Lebesgue decomposition: $\lambda=\lambda_a+\lambda_s$ where $\lambda_a ...
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145 views

Riemann or Lebesgue integrable

Lets consider the following two functions: $$f(x)=\begin{cases} x &,x\in[0,1]\setminus\Bbb Q \\ 0 &,x\in[0,1]\cap\Bbb Q\end{cases}$$ $$g(x)=\frac{(-1)^{[x]}}{[x]}$$ where $[x]$ is the ...
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102 views

Show that a function is continuous

Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$. Show that the function $g$ defined on $\mathbb{R}$ by $g(t) = ...
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205 views

Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
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464 views

Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable

Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
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55 views

Is this function measurable? Something to do with Bochner space and norms.

Suppose $f:[0,T]\to X$ is a measurable map where $X$ is Hilbert space. Suppose also that $R(t):X \to X^*$ is an isometric isomorphism with $$\lVert R(t)f(t)\rVert_{X^*} = \lVert f(t) \rVert_X$$ also ...
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103 views

Measure of the set of real numbers that can be approximated in this way

Let $$A = \{x \in \mathbb{R}\mid \exists\,\text{infinitely many pairs of integers $p,q$ such that $|x-p/q| \leq 1/q^3$}\}.$$ Is the measure of $A$ equal to $0$? Any ideas?
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96 views

Two random variable with the same variance and mean

Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely. My partial answer: By the definition of conditional expectation we have ...
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49 views

Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?

Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in ...
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71 views

Showing that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}[\exp[tX]]$

The problem is to show that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}(\exp[tX])$ given $\exp(tX)<\infty$ for $t\in \mathbb{R}$ where $X$ is a random variable. Then to show that ...
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59 views

$\int_\Omega \phi d\mu_n\to\int_\Omega\phi d\mu,\forall\ \phi\in C_0(\Omega) $ and $\mu_n\geq 0$ implies $\mu\geq 0$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and suppose that $\mu_n$ is a sequence of non-negative Radon measures, that converges to a Radon measure $\mu$ in the weak star sense: $$\int_\Omega ...
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31 views

If $\int_a^b u(t)v(t)$ exists for every $v \in C_0^\infty(a,b)$, does $\int_a^b u(t)$ exist?

If $\int_a^b u(t)v(t)$ exists for every $v \in C_0^\infty(a,b)$, does $\int_a^b u(t)$ exist? Is $u$ measurable? I think yes since we can approximate the function $1$ by infinitely differentiable ...