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

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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 ...
6
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1answer
1k views

Closure, Interior, and Boundary of Jordan Measurable Sets.

This question has a number of parts. Let $E\subset\mathbb{R}^{d}$ be a bounded subset. (1) Show that $m^{\star,(J)}(E)=m^{\star,(J)}(\bar{E})$ (closure) (2) Show that ...
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2answers
108 views
+400

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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2answers
1k views

Uniform integrability and tightness.

Definition: Let $(X,M,\mu)$ be a measure space and $\{f_n\}$ a sequence of measurable functions on $x$ that are integrable. Then $\{f_n\}$ is uniformly integrable if for every $\epsilon >0$, there ...
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2answers
37 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 ...
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1answer
38 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 ...
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2answers
82 views
+100

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 ...
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2answers
55 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
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1answer
19 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 ...
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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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1answer
30 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 ...
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1answer
172 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. ...
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0answers
19 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
41 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 ...
5
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2answers
127 views

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$?

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$? Think when we look at the equivalence classes of almost ...
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0answers
36 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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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..
2
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2answers
3k views

Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$

I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
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1answer
44 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 $n$ ...
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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 ...
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2answers
270 views

Probability space for stochastic processes

In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space ...
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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 ...
2
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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 ...
7
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2answers
108 views

If $A_1 \subset A_2 \subset \mathbb R$ and $m^*(A_1) = m^*(A_2)$, will $m^*(A_1 \cap T) = m^*(A_2 \cap T), \forall T \subset \mathbb R$?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} l(I_n): E \subset ...
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1answer
19 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 ...
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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 ...
4
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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 ...
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2answers
78 views

Condition implying tightness of sequence of probability measures

A sequence of probability measures $\mu_n$ is said to be tight if for each $\epsilon$ there exists a finite interval $(a,b]$ such that $\mu((a,b])>1-\epsilon$ For all $n$. With this information, ...
1
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1answer
53 views

Abstract enunciation of the Good Set Principle in measure theory

I am struggling with the Good Set Principle in Measure Theory, so is this rephrasing in the most abstract way ultimately correct? Good Set Principle Let $(X, \Sigma)$ be a measurable space. ...
4
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1answer
44 views

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...
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2answers
413 views

$\sigma$-finite measure and $\sigma$-semi-finite measure

Let $ (X, \Sigma, \mu) $ it will be a space with measure. $\mu$ is $\sigma$-finite measure if it exist sequence of sets $X_{i} \in \Sigma $ and $\cup_{i=1}^{\infty}X_{i}=X$ and ...
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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 ...
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0answers
19 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 ...
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2answers
143 views

Lang's treatment of product of Radon measures

Let $X$ be a locally compact Hausdorff space. We denote by $\mathcal B(X)$ the $\sigma$-algebra of Borel sets of $X$. A positive Radon measure $\mu$ on $X$ is a measure defined on $\mathcal B(X)$ with ...
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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 ...
3
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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 ...
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$\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, $\mu_{\mathcal{C}}-$a.e

If $A \subseteq \mathcal{C}$ is $\mu_{\mathcal{C}}$-measurable, then $\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, ...
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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 ...
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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 ...
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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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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 ...
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0answers
29 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 ...
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0answers
28 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 ...
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2answers
2k views

Steinhaus theorem (sums version)

This is a question from Stromberg related to Steinhaus' Theorem: If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval. I can't quite see how to modify the ...
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0answers
91 views

Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} p_n = 1$. Define P: $\mathfrak{F} \to ...
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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
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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 ...
30
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2answers
9k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
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 ...
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2answers
119 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 ...