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

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2
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2answers
59 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
1
vote
2answers
42 views

Does construction of infinite product measure require axiom of choice?

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...
0
votes
1answer
31 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
0
votes
1answer
26 views

Question about the definition of convergence in measure.

In my text book convergence in measure ${{f}_{k}}\mathop \to \limits^{m} {f}$ is defined as "$\forall \epsilon> 0$ we have $\mathop {\lim }\limits_{k \to \infty } |\{ x \in \Omega :\left| ...
1
vote
2answers
31 views

Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Let $f_n$ is a sequence of measurable functions on a measure space $(X,\mathcal{B},m)$ converging pointwise to a function $f$. Suppose that $f_n$ is integrable for all $n$ and ...
1
vote
2answers
41 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
3
votes
1answer
50 views

Existence of Lebesgue measure on real line proof help

I am reading a proof of the existence of Lebesgue measure and am struggling to understand one part. I will first get you up to where I am in the proof. We define for a set written as a finite ...
2
votes
1answer
22 views

Bernoulli product measure

Let $\Omega=\{0,1\}^\mathbb{N}$ and $\mathcal{A}$ the sigma-algebra generated by the cylinders sets $\{w\in\Omega\vert \forall s \in S, w_s=\epsilon_s\}$ with $S\subset\mathbb{N}$ finite and ...
2
votes
2answers
56 views

Measures on $\mathbb{R}$ that are not translation invariant

I am looking for examples of measures on $\mathbb{R}$ which are not translation invariant. The only one I could come up so far is the dirac measure. In particular, I am looking for a measure $\mu$ ...
2
votes
1answer
35 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
1
vote
1answer
31 views

Doubling measure of an annulus

Recall that a doubling measure is a measure with the additional requirement that: $$\mu(B_{2R})\le C_\mu \mu(B_R)$$ for some contstant $C_\mu$. While solving some esercises related to doubling ...
2
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0answers
22 views

The dual space of weighted compact supported function?

Let $\Omega\subset \mathbb R^N$ be open bounded. It is well know that the dual space of $C_c(\Omega)$, i.e., compacted supported continuous function, can be identified by finite Radon measure ...
2
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0answers
44 views

Question on Egoroff-like theorem

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...
2
votes
0answers
40 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
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votes
0answers
18 views

Corresondance of measures and functions.

Are there situations other then the Reimann-Stiljtes integal where this correspondance is important/useful? I cant come up with any..
1
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1answer
45 views

How to use induction to show that $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n) $ are independent?

I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ ...
1
vote
2answers
18 views

Sigma algebra definition and Lebesgue integration

I know the definition of the $\sigma$-algebra, and I have seen it used in integration theory. However, I do not understand why it is defined the way it is. From what I understand, the definition ...
0
votes
1answer
178 views

Is $e^x$ finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $?

Does $e^x$ is finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $? I think it is, but I put on this question to make sure. I know $f$ being finite a.e. ...
2
votes
2answers
33 views

Showing that a Borel Measure $\mu\equiv 0$

Problem. Let $\mu$ be a Borel measure on $[0,1]$. Assume that $\mu$ and Lebesgue measure $m$ are mutually singular. $\mu([0,t])$ depends continuously on $t$. $f\in L^{1}(\mu)$ for any ...
1
vote
1answer
54 views

Is proving $m(E) < \epsilon, \forall \epsilon > 0$ equivalent to prove $m(E) = 0$?

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function ...
0
votes
1answer
20 views

Question related to the construction of product measure

I am learning about product measures and I was stuck on a detail of the proof. I would appreciate any assistance! Suppose we have a measure spaces $(X_i, M_i, \mu_i), i=1, ..., N$, that are complete ...
0
votes
1answer
29 views

if the integrals of a non-negative sequence of functions go to zero, does this imply functions go to zero a.e.? [duplicate]

This question arised when I was dealing with an old qual problem, and if this is true, I'll be done, but I'm not sure if it's true or not: Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of ...
3
votes
5answers
121 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
-1
votes
0answers
11 views

$\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, $\mu_{\mathcal{C}}-$a.e

If $A \subseteq \mathcal{C}$ is $\mu_{\mathcal{C}}$-measurable, then $\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, ...
1
vote
2answers
84 views

Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
0
votes
0answers
26 views

The relationship between outer measures and smallest coverings

Recall that if $\mathcal{A}\subset \mathcal{P}(X)$ is an algebra and $\mu_{0}:\mathcal{A} \to [0,\infty]$ is a premeasure on $\mathcal{A}$ then we can define the outer measure $\mu^{*}$ for any set ...
0
votes
1answer
43 views

Understanding Cohn's Radon-Nikodym proof from his book on measure theory

The part of the proof which I don't get is $$\nu(A)=\int_{A} g\ \mathsf d\mu$$ where $g$ is Radon-Nikodym derivative. He has a set of functions for which $$\int_{A} f\ \mathsf dx \le \nu(A) ,$$ he has ...
3
votes
0answers
41 views

Generalization of the Vitali-Hahn-Saks Theorem

Is there a generalization of the Vitali-Hahn-Saks Theorem for nets of measures? I do not find any related literature. Take a sequence of bounded measures on a sigma-field and consider a subnet of this ...
2
votes
0answers
30 views

Is there a programmatic way to calculate cascaded sigma functions?

Let my format be sigma(function,from,to) = f(n) for example sigma(sigma(1 , j = 1 , j = i) , i = 1 , i = n) = (n^2)/2 ...
2
votes
0answers
20 views

How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
1
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0answers
29 views

Expectation of a continuous function

Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and ...
4
votes
1answer
64 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...
1
vote
1answer
23 views

Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
1
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1answer
23 views

Question on mutual singularity and absolute continuity of complex measures

I was presented these two somewhat similar questions from Folland's real analysis (second edition) dealing with complex measures and their mutual singularity and absolute continuity. They are 3.19 and ...
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votes
0answers
16 views

Behaviour of Fourier Inverse Transform after non-linear modulation

Suppose $\phi$ is a continuous nowhere differentiable function. $g$ some function in Schwartz space such that $\hat{g}$ has compact support. Define $f(x) = \int_{-\infty}^\infty ...
1
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1answer
30 views

A subset on which a measure is strictly positive

There is a "cake" which is represented by the interval $[0,1]$. There is a non-atomic value measure $V$ defined on the cake. I would like to define an algorithm for dividing the cake between two ...
0
votes
1answer
58 views

Find Lebesgue measure of $\limsup A_n \cap B_n$ if $m(\limsup A_n)=m(\liminf B_n)=1$,

Let $m$ be the lebesgue measure on $X=[0,1]$. if $m(\limsup\limits_{n\rightarrow{\infty}} {A_n})=1$ and $m(\liminf\limits_{n\rightarrow{\infty}} {B_n})=1$, prove that ...
3
votes
1answer
26 views

Difference between a measure and a premeasure

I am new to measure theory and am wondering: Is the only difference between a measure and a premeasure the fact that measures are defined on $\sigma$-algebras and premeasures are defined only on ...
2
votes
1answer
23 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
9
votes
3answers
806 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
2
votes
2answers
36 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
0
votes
1answer
37 views

$\limsup = \liminf$ of sequence of Sets

This problem was on my in-class final for a measure theory course I took in the fall, and now I am studying for my qualifying exam so I am trying to figure this one out: Suppose ...
1
vote
1answer
21 views

Does convergence in $L_p$ imply convergence of quotients in $L_p$

Take the measure space to be $\mathbb{R}$ with Borel $\sigma$-algebra and Lebesgue measure (Although just thinking in terms of a general measure space probably works for this problem.) Question: True ...
1
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1answer
51 views

Definition of upper integral

Answering this question it occurred to me that the OP's definition of integral is unsatisfactory in the following sense. He defines it using the usual Lebesgue integral. I think it would be far more ...
1
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2answers
34 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
9
votes
2answers
121 views

Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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0answers
45 views

A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
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1answer
28 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
1
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1answer
24 views

What is the countable product sigma algebra of powersets of a countable set $E$? The powerset of the space of all sequences in $E$ or not?

Let $E$ be a countable set with power set $\mathcal{P}(E)$. $(E,\mathcal{P}(E))$ is a measurable space. Let $E^{\mathbb{N}}$ be the space of sequences in $E$ and $\mathcal{P}(E)^{\mathbb{N}}$ the ...
0
votes
1answer
17 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...