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

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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
46 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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0answers
14 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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2answers
17 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
15 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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27 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
46 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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18 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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40 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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54 views

Measurability properties of processes that arise as limits of sequences of measurable processes

I try to reduce my problem to a more general statement from which I want to know whether this is true in general. I have a sequence of continuous-time stochastic processes $X_t^{(n)}, t \geq 0$ with ...
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21 views

If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$.

If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$. Is it possible to prove this theorem?
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2answers
46 views

almost every where property

I have question ,let $f$ is measurable and integrable i.e $f\in(\Omega,\mathbf{A}, \mathbb{R})$ and for all $A\in \mathbf{A}$ $\int_Af(\omega)d\mu=0$ show that $f=0$ almost every where. answer: I ...
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1answer
40 views

Inequality with outer measure

Let $ \mu : \mathcal{H} \to \mathbb{R} $ be a measure on a seminring $ \mathcal{H} $ over a set $ X $ and $ \mu^{*} = \inf \{ \sum \mu(A_{i}) : A_{i} \in \mathcal{H}, \; A \subset \bigcup A_{i} \} $ ...
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1answer
46 views

Hardy–Littlewood maximal function – an example

Let $$f(x)= \begin{cases} \frac{1}{x\log^2x}, & \text{if} \hspace{2mm} 0 < x < \frac{1}{2}\\ 0, & \text{otherwise} \end{cases} $$ I have so far shown that $f$ is integrable. However, I ...
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21 views

Lebesgue outer measure of linear subspace

Prove that $\lambda^{*}(A)=0$, where $\lambda^{*}$ is $n$-dimensional Lebesgue measure on $\mathbb{R}^n$ and $A$ is $k$-dimensional subspace of $\mathbb{R}^n$ and $k<n$. I've proved this for ...
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1answer
26 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
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38 views

Which projection, in $L_\infty$ norm or $L_2$ norm, is non-expansion?

I am just wondering which projection is non-expansion? Basically, I am wondering if $F$ is a projection operator then which norm would satisfy the following non-expansion property, where for a given ...
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110 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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46 views

Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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28 views

$\mu_x\otimes\mu_y$ and $\mu_y\otimes\mu_x$

Let $X$ and $Y$ spaces endowed with measures $\mu_x$ and $\mu_y$ defined on set semirings $\mathfrak{S}_x$ and $\mathfrak{S}_y$ and let $A\subset X\times Y$ be a subset of $X\times Y$ such that ...
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26 views

approximation simple functions with finite support

Let $f$ be a nonnegative measurable function. I want to prove that there is an increasing sequence of nonnegative simple functions each of which vanishes outside a set of finite measure such that ...
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1answer
21 views

What does $\mathbb{R}$-invariant mean for a measure?

Let $(X, A, m)$ be a measure space with m being $\mathbb{R}$-invariant. What does this mean?
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1answer
46 views

Proof of Outer Regularity of Lebesgue Measure on R

Let $E \subseteq \mathbb{R}$ be a measurable set, and $\epsilon > 0 $. Show that there is an open set $G \supseteq E$ such that $\mu(G \setminus E) < \epsilon$. Any hints? By the definition of ...
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27 views

Does Fubini's theorem imply $\int_X (\int _Y f _x d \lambda )d \mu=\int _X d \mu \int _Yf(x,y) d \lambda$?

I need some help with intepretating the result of Fubini's theorem. define $ \phi (x) =\int_Y f _x d \lambda $ and $ \psi (y)= \int _X f _y d \mu$ According to Rudin, Fubinis theorem tells us that ...
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38 views

A Question on Lebesgue Dominated Convergence Theorem

I have a general question about the dominated convergence theorem. The theorem states that if I have a sequence of measurable functions that are bounded by an integrable function and converge ...
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1answer
22 views

What does $\lVert\mu \rVert=1$ mean for a measure $\mu$ on a compact metric space?

What does $\lVert\mu \rVert=1$ mean for a measure $\mu$ on a compact metric space $\Omega$? Sorry, I would like to add some own ideas, but I do not have... it has to be a kind of normalization on ...
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36 views

Functional representation of adapted jointly measurable stochastic processes

Let $X_t : \Omega \to E, \ t \geq 0$ be continuous-time stochastic process with (Polish) state space $E$ and canonical filtration $\mathcal{F}_t := \sigma(X_u \ | \ u \leq t)$. Let $Y_t : \Omega \to ...
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1answer
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$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
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31 views

On Compact and Measurable Sets with Positive Length

Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure ...
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40 views

Proving that any measure is the sum of a semi-finite measure and a measure which takes either 0 or infinity.

I need help proving that for a measure space (X, A, u) that u can be written as the sum of a semi-finite measure and a measure that takes on the values 0 and infinity. The second measure, u_i I ...
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1answer
45 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
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56 views

Is there a dense set of positive measure which does not contain any open set?

I want to construct $A\subseteq[0,1]$ with $m(A)>0$ and for every open subset $U$ of $[0,1]$, $0<m(A\cap U)<m(U)$ and $U\not\subseteq A$. I think these sets must have measure zero. ...
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3answers
43 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
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29 views

Product of counting measure and the integral

Given the sigma algebra $P(\mathbb{N}^2)$(or $P(\mathbb{Z}^2)$ and counting measure $n$, I need to show that $n \times n$ is the counting measure for the aforementioned sigma algebra and compute the ...
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26 views

Existence of a A measurable function

Let $A$ be sigma algebra having subsets of $R$ only. We define a function from subset of $A$ to $R$ is said to be $A$ measurable iff every Borel set is pulled back to elements of $A$. Is there a ...
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39 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
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1answer
39 views

necessary conditions of measure approximation theorem

Measure approximation theorem (I can't really remember its exact name) states that let $A$ be an algebra, $\mu$ a measure on $\sigma(A)$ and $\mu$ is $\sigma$-finite on $A$. Let $E\in \sigma(A)$ such ...
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1answer
42 views

Sequence of functions divided by constants converge to zero

I am trying to show the following: If $\{ f_n \}$ is a sequence of a.e. real-valued measurable functions in X, and the measure $\mu(X) < \infty$, there exist positive constants $a_n$ such that ...
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72 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...
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38 views

Showing that set contains no intervals.

Hi, I'm trying to solve a problem: Here goes: First part is true because sets $B$, $B'$, and $E$ are countable and hence $F$ is countable so it's Lebesgue measure is $0$, thus it contains no ...
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36 views

Is this proof correct()?

Prove that for $p=\infty$ : $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies f=0 $ a.e on $\Omega$ Proof $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies ess$ $\sup |f| =0 \implies \inf\{a\in R| ...
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1answer
58 views

Suppose $A_n \supset A_{n+1}$ . Show $P(A_n) \searrow P(\cap_{n=1}^\infty A_n)$

I know the proof for if it is a monotonic increasing sequence of sets using the fact that $\sum_{i=1}^\infty P(B_i)$ goes to 0, where $B_i$ is a partition of ${A_i}$.
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60 views

Linearity of the integral without $\sigma$-additive measures

I was wondering how you could prove the linearity of the integral without using that measures are $\sigma$-additive. I have no clue of where to start, but let me state my question more precisely. ...
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Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
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Extending a pre-measure on an sigma-algebra?

Consider on $\Bbb{R}$ the family $\Sigma $ of all Borel sets which are symmetric w.r.t. the origin, which is a $\sigma $-algebra. Is it possible to extend a pre-measure $\mu $ on $\Sigma $ to a ...
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1answer
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Modification of Set Function in Construction of Lebesgue Measure

Suppose in the construction of Lebesgue measure we replace the set function $\mu((a,b))=b-a$ with $\mu((a,b))=\sqrt{b-a}$. What can we say about $\mu^*$ and the $\sigma$-algebra of measurable sets? ...
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Is sigma-finiteness required?

Suppose that f and g are two extended-real valued measurable functions on an arbitrary measure space, such that the integral of g dominates that of f on every set A of the sigma-field. Is it true ...
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57 views

Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory

I'm taking stochastic probability class but I'm now only taking analysis (with Rudin's PMA) class. The stochastic probability class doesn't depend heavily on the theoretic structures: rather, the ...
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35 views

Reciprocal of measurable function is measurable

Let $f(x)$ be a measurable function and define $$g(x)= \begin{cases} \frac{1}{f(x)}, & f(x) \not= 0 \\ 0, & f(x)=0 \\ \end{cases} $$ Show that $g(x)$ is also measurable. Here's my reasoning ...
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32 views

Measure theory mapping sets to groups?

This is a question from a physicist wondering if a certain idea in mathematics has been developed. Intuitively, suppose I have a number of objects distributed in space. I want a function that given a ...