Tagged Questions

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

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Borel Measures: Discrete Decomposition

Context The notion of atoms and point masses agree to certain extent. (See Summary on Atoms.) Measures decompose w.r.t. atoms. (See Paper on Atoms.) Here, the goal is a direct approach to decompose ...
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1answer
19 views

Is a measure, which is equivalent to a discrete measure, also discrete?

Let $(\Omega,\mathcal F)$ be a measurable space. Define a probability measure by $\mathbb P=\sum_{k=1}^\infty\alpha_k\delta_{\omega_k},$ where $(\omega_k)_{k\in\mathbb N}\subseteq \Omega,$ ...
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0answers
9 views

Integrable equivalence in $\sigma$-finite measure space (Proposition from Bogachev)

I'm reading Bogachev's measure theory book und I don't understand a proof from a proposition in this book. Maybe someone could help me... Here is the statement of Proposition 2.6.2 (ii): Let $(X, ...
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1answer
6 views

Does the pushforward operator (on measures) preserve surjectiveness?

Let $I = [0,1]$ be the unit interval. Let $\pi: I \to I$ be a Borel-measurable surjective map. Is the pushforward operator $\pi_*: \mathcal P(I) \to \mathcal P(I)$ surjective as well, where $\mathcal ...
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2answers
30 views

Closed sets with empty interior measure zero

Is the Lebesgue measure of a closed set with empty interior in $\mathbb{R}^{n}$ always zero? Trying to understand something in the math notes that I don't understand, and if the above is true, it ...
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1answer
14 views

How do I see formaly that $\liminf_{n \rightarrow \infty} |f_n|^p = |f|^p$ for $p \in [1, \infty)$?

Let $(f_n)_{n \in \mathbb N}$ be a sequence of functions s.t $\lim_{n \rightarrow \infty} f_n = f$, where $f_n : X \rightarrow \mathbb R$. How do I see formaly that $\liminf_{n \rightarrow \infty} ...
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1answer
27 views

Boundedness of sequence of functions

Consider a sequence of continuous integrable functions $\{f_n(t)\}_n$ such that $\ast\ \displaystyle\lim_{n\rightarrow\infty}f_n(t)=0$ for all $t>0$ $\ast\ \{f_n(t)\}_n$ is such that ...
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Find the derivative of $F(t) = \int_{0}^{\infty} \cos(t^2 h(x))e^{-x} \lambda(dx) \ \ (t \in \mathbb R)$ using derivative theorem.

Let $(X, \mathcal E, \mu)$ be a measure-space and $I \subseteq \mathbb R$ an open interval. Let $f: X \times I \rightarrow \mathbb R$ be a function. Consider $$F(t) = \int_X f^t(x) \mu(dx)=\int_X ...
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2answers
23 views

Functions in $\mathbb{R}^{d}$ which are integrable or non-integrable (Stein & Shakarchi)

I'm reading Stein and Shakarchi's book on real analysis, and twice already they've mentioned the following without proving it: $$f_{a}(x) = \begin{cases}|x|^{-a} & |x| \leq 1 \\ 0 & |x| ...
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1answer
45 views

If $f$ is continuous, then $\lim\limits_{n \rightarrow \infty} \int^b_a n(f(x+ 1/n)-f(x)) \lambda(dx) = f(b)-f(a)$

Consider a continuous function $f: \mathbb R \rightarrow \mathbb R$ and define $f_n: \mathbb R \rightarrow \mathbb R$ by $f_n(x) = n(f(x+1/n)-f(x))$. I want to show that for $a < b \in \mathbb R$ ...
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0answers
15 views

Central Limit Theorem for transformed random variables

The Central limit theorem (CTL) is often given similar to the entry in Wikipedia as: Suppose ${X_1, X_2, ...}$ is a sequence of independent and identically distributed random variables with ...
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1answer
33 views

A sequence of truncates of $f$

If $f$ is measurable and $A>0$ then the truncation $f_{A}$ defined by: $$f_{A}(x)=\begin{cases} f(x)&\text{if $\left | f(x) \right |\leq A$}\\ A&\text{if $ f(x)> A ...
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1answer
17 views

Why do only open cubes shrink regularly?

$\textbf{DEFN}$ A collection of sets $\{U_{\alpha}\}$ is said to shrink regularly to $\overline x$, if there is a constant $c>0$, such that for each $U_{\alpha}$ there is a ball $B$ with ...
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1answer
30 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
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0answers
8 views

Total variation of sum of measures

I'm working on proving that, given two signed measures $\nu_1$ and $\nu_2$ on $(X,M)$ that both omit either $\infty$ or $-\infty$, $|\nu_1 + \nu_2| \leq |\nu_1|+|\nu_2|$ using the definition $|\nu| = ...
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1answer
15 views

Do outer measures satisfy all the properties of a measure?

I know the definition of a measure and an outer measure; however, does an outer measure by default also fulfill the properties of a measure?
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2answers
55 views

Finding functions with $\phi (\lim_{p \to 0}||f||_p)=\int_{0}^1 (\phi \circ f)dm$

If $m$ is Lebesgue measure on $[0,1]$ , for what functions $\phi$ on $[0,\infty)$ does the relation $$\phi (\lim_{p \to 0}||f||_p)=\int_{0}^1 (\phi \circ f)dm$$ hold for any bounded, measurable ...
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1answer
55 views

integral over $\sin(x)/(x^p)$ from 0 to $\infty$ [on hold]

I have a question about the convergence of $\int^\infty_0 \frac{\sin(x)}{x^p} dx$ for $p\in\mathbb{R}$ what can I say for convergence of this function ?? I know that $\int^\infty_0 ...
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1answer
23 views

The exist $B$ s.t. $A \subset B$ and s.t. $\mu(B) \le \mu(A)+\epsilon$ for any $\epsilon>0.$ [on hold]

How to show the following is true. The exist $B$ s.t. $A \subset B$ and s.t. $\mu(B) \le \mu(A)+\epsilon$ for any $\epsilon>0.$
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1answer
30 views

Measurability of derivative of Lebesgue integral function

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа that if $f:[a,b]\to\mathbb{R}$ is a Lebesgue-summable function on its domain then the derivative $\Phi'$ of the integral ...
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1answer
19 views

Completion of sigma field

Suppose $(\Omega, \mathcal{A},P)$ is a probability space. Show that \begin{align} \mathcal{A}^*=\{A \cup N: A \in \mathcal{A}, N \in \mathcal{N} \} \end{align} is a sigma algebra and where ...
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0answers
7 views

Area Calculation: Equivalence between Riemann Integral and Completing Square by translation

If we want to calculate area of an triangle or equivalently want to calcuate area under curve $y=x$ ($x \in [0, 1]$), we can use two methods: 1)- Assuming we know the area of the square, we can ...
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2answers
14 views

Let $\phi: I \rightarrow \mathbb R$ a convex function. Show $\phi$ is $\mathcal B(\mathbb R)_I$-$\mathcal B(\mathbb R)$-measurable.

Let $I \subset \mathbb R$ be an interval and $\phi: I \rightarrow \mathbb R$ a convex function. Show $\phi$ is $\mathcal B(\mathbb R)_I$-$\mathcal B(\mathbb R)$-measurable. I know that if $\phi$ ...
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1answer
13 views

Generating set for $\sigma(\mathcal{G}, X)$ where $\mathcal{G}$ is sub sigma field and X is a r.v.

I'm trying to prove the following fact. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-field and let $X : (\Omega,\mathcal{F},\mathcal{P}) \rightarrow (S,\mathcal{S})$ be a random variable. ...
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0answers
15 views

Prove Tonelli and Fubinis theorems imply results regarding summation-order of $\sum \sum a_{m,n}$ where $a_{m,n} > 0$.

Prove Tonelli and Fubinis theorems imply results regarding summation-order of $\sum \sum a_{m,n}$ where $a_{m,n} > 0$. I've already proved that $\tau_2 = \tau_1 \otimes \tau_1$ where $\tau_i$ ...
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1answer
45 views

Are these outer measures?

Are the following outer measures? (i) $\mu^*(A)=0$ if $A \subseteq X$ countable and $\mu^*(A)=1$ otherwise. (ii) $\mu^*(A)=0$ if $\{x,y\} \cap A = \emptyset$ and $\mu^*(A)=1$ otherwise. (iii) ...
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1answer
20 views

Fubini theorem to calculate iterated integral, counting and Lebesgue measure

With the counting measure on $\mathbb{N}$, and the Lebesgue measure on $\mathbb{R}$, consider their product measure space and the function: $f(x,n) = \begin{cases} -2^n & \text{ if ...
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1answer
24 views

$f$ and $g$ are equal almost everywhere

I'm looking for an example of two functions that are equal almost everywhere, one of them is measurable but the other one is not (I think this is only possibly in a not-complete measure space). I ...
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1answer
43 views

Does uniform convergence imply $L^1$-convergence?

Suppose that $(X,M,\mu)$ is a measure space. Is it true that if $\,f_n \in L^1(X)\,$ and $\,f_n\,$ converges uniformly to $f$, then $\,f \in L^1(X)\,$ and $\,\int_X f_n\,d\mu\,$ converges to ...
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1answer
40 views

Do all the open sets in real space have measure nonzero?

$\mu$ is the Lebesgue measure on $\mathbb{R}^n$, A is a null set (ie $\mu(A)=0$). If $\mathbb{R}^n\backslash A$ is closed, does it imply $A=\emptyset$? What happens in general? Different measure in ...
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1answer
23 views

Is the composition of a measurable function with a monotone function measurable?

Assume that $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is a strictly monotonically increasing function. Is it true that a real valued function $f:X\rightarrow\mathbb{R}$ is measurable on $(X,M)$ iff its ...
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0answers
23 views

dominated convergence function for log(x)

what is dominating function for f(x)=log(x) or f(x)=log (x-1),I mean what is suitable integrablre g(x) such that $\mid f(x)\mid<g(x)$
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1answer
25 views

Are the following outer measures?

I have to check, if the following is an outer measures and if so, then determine the corresponding $\sigma$-algebra $\mathcal{A}(\mu^*)=\{A \subseteq X: A \text{ is } \mu^*\text{ - measurable }\}$. ...
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1answer
19 views

For$ f$ non-negative function, show that $f$ is meseurable if for some $p>0$ $f^p$ is measurable.

For$ f$ non-negative function ($f \geq 0$), show that $f$ is meseurable if for some $p>0$ the function $f^p$ is measurable.
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2answers
49 views

Does equality of antiderivatives imply equality almost everywhere?

If two Lebesgue integrable functions $\,f,g:[a,b]\to \mathbb R\,$ satisfy $$ \int_a^x f(s)\,ds= \int_a^xg(s)\,ds, $$ for every $\, x\in[a,b],\,$ then is is it true that $ f(x)=g(x)$ almost ...
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1answer
25 views

Find all measurable sets whose subsets are measurable

Find all Lebesgue measurable sets $A ⊂ \mathbb{R}$ with the following property: All subsets $B ⊂ A$ are measurable.
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1answer
32 views

Probability Theory: Conditional Independence and Independence

I have the following definition of conditional independence: $X$ and $Y$ are called conditionally independent given a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$ if for all bounded Borel ...
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0answers
20 views

How to define probability density function in Hilbert space??

Consider the space of random continuous functions $f:(0,1)\rightarrow\mathbb R$. Suppose we assume that this is a Hilbert space. Is there any notion of probability density function in the Hilbert ...
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1answer
26 views

Counterexample: if $f_n\rightarrow f$ in measure then $\frac{1}{f_n}\rightarrow \frac{1}{f}$ in measure

I was trying to find an example showing that this statement does not always hold. If a sequence of positive measurable function $f_n$ converges to $f$ in measure then $\frac{1}{f_n}$ converges to ...
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0answers
31 views

Why does Tonellis theorem imply that the order of integration doesn't matter?

Let $(X, \mathcal E, \mu)$ and $(Y, \mathcal F, \nu)$ be measure spaces. In my textbook Tonellis theorem is stated for a non-negative function $f \in \mathcal M(\mathcal E \otimes\mathcal F)^+$ s.t ...
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2answers
34 views

Generators of the Borel-$\sigma$-algebra on $\mathbb{R}$

Prove that the following sets generate the Borel-$\sigma$-algebra $B(\mathbb{R})$ $$X_1 = \{ (-\infty,a), \, a \in \mathbb{R} \} \\X_2 = \{ (-\infty,b], \, b \in \mathbb{R} \}$$ We had defined ...
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0answers
15 views

Proof of Egoroves Theorem

Almost all books and articles on measure theory give proof of Egorove's theorem by considering a sequence of double-indexed measurable sets (see this, for example). I don't find any natural way to ...
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1answer
37 views

Measurable Functions as Limits (a.e) of Step Functions

I saw the following theorem and its proof. Theorem 1: Given a measurable function $f$ on $E\subseteq \mathbb{R}$, there exists a sequence of simple functions $\{f_k\}$ which converges point-wise to ...
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1answer
21 views

All right or left continuous processes are jointly measurable

I was reading the Stochastic Calculus notes by George on this website http://almostsure.wordpress.com/2009/11/03/stochastic-processes-indistinguishability-and-modifications/ and I cannot understand ...
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1answer
47 views

Differentiation under the integral sign and counting measure

Consider a power series $f(x)=\sum_{n=1}^\infty a_nx^n$, and assume that $\displaystyle R=\lim_{n\to \infty} \frac{a_n}{a_{n+1}}$ exists. Use differentiation under the integral sign to show that ...
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0answers
36 views

Differentiation under the integral sign for definite integrals [on hold]

Use Differentiation under the integral sign to compute $\displaystyle \int_0^\infty \frac{dx}{(x^2+p)^n}$ for any $n\in\mathbb{N}$. No idea even to start the solution. Can anyone please give me a ...
2
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1answer
33 views

Integration of complex functions

Show that $\displaystyle \int_{-\infty}^\infty x^{2n}e^{-x^2}dx=(2n)!\frac{\sqrt\pi}{4^n n!}$ by differentiating the equation $\displaystyle \int_{-\infty}^\infty e^{-tx^2} dx=\sqrt{\frac{\pi}{t}}$. ...
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0answers
20 views

Lebesgue differentiation theorem implies Lebesgue density.

I am asked to prove that for every $ E \subset \mathbb{R}^d$ there exists some cube $Q$ such that $$m(E \cap Q) \ge (1 - \epsilon) m (Q)$$ I am fairly confident I know how to complete this problem. ...
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1answer
34 views

A compact set $K \subset \mathbb{R}$ of positive Lebesgue measure such that $m(K \cap I) < |I|$ for every interval $I$ of positive length

There exists a compact set $K \subset \mathbb{R}$ of positive Lebesgue measure such that $m(K \cap I) < |I|$ for every interval $I$ of positive length For this question, I am thinking about ...
0
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1answer
36 views

Finest Measurable Partition

Disclaimer: This question is part of: Borel Measures: Atoms (Summary) Given a sigma algebra $\Sigma$ over a countable space $\#\Omega\leq\aleph_0$. Does it admit a finest measurable partition: ...