Measure theory Definition and 136 Threads

Dyadic cubes of order ##n\in\mathbb N## are sets of the form $$C=\prod_{j=1}^d (k_j2^{-n},(k_j+1)2^{-n}],\quad k_j\in\mathbb Z.$$ I know that the dyadic cubes in ##\mathbb R^d## induce a generating collection for the Borel ##\sigma##-algebra of ##U## (by intersecting the dyadic cubes in...

I quote; I struggle with the last sentence and putting the pieces together. I know the abstract change of variables formula for ##\nu## on ##(F,\mathcal D)## being the pushforward measure of ##\mu## on ##(E,\mathcal C)## under ##\varphi:E\to F##, i.e. ##\nu(A)=\mu(\varphi^{-1}(A))## for every...

##L^2##-space is defined as equivalence classes on the set ##\mathcal L^2## of squared integrable measurable functions ##f## defined on the measure space ##(\Omega, \mathcal A, \mu)##. The equivalence relation ##\sim## is: ##f \sim g## iff ##f=g## almost everywhere (a.e.). Prove that the above...

In the Riemann-Lebesgue lemma, the author says it suffices to prove $$\hat{f}(\xi)\underset{|\xi|\to\infty}{\to}0$$for step functions on ##\mathbb R## only (step functions are simple functions where the sets of the indicator functions are intervals in ##\mathbb R##). This is because the step...

I fail to see why the "by-product" of Proposition 4.6 is true. Isn't $$f_n\stackrel{L^p}{\to} f \text{ and } f_n(x)\to g(x) \ \mu(\mathrm{d}x)\text{-a.e.}$$ the same statement? I am confused about how to apply Proposition 4.6 to prove the "by-product" statement. Any help is greatly appreciated.

Let ##(E,\mathcal A)## be a measurable space equipped with a measure ##\mu##. If ##f:E\to\mathbb R## is integrable, then we have ##\left|\int f\,\mathrm{d}\mu\right|\leq\int |f|\,\mathrm{d}\mu##. If ##f:E\to\mathbb C## is integrable, Le Gall in his book Measure Theory, Probability and Stochastic...

Let ##(f_n)## be a sequence of measurable functions from ##E## into ##\mathbb R##. I'm reading a proof of the fact that the set ##A## of all ##x\in E## for which ##f_n(x)## converges in ##\mathbb R## as ##n\to\infty## is measurable. The proof goes like this (I'm paraphrasing): Why is...

Let ##m## be Lebesgue measure. It is another proposition that the functions ##NBV## are in one-to-one correspondence between complex Borel measures, e.g. ##F\in NBV## induces a complex measure ##\mu_F## such that ##F(x)=\mu_F((-\infty,x])##. Then in Folland's real analysis text, I'll omit the...

I'm reading a proof of a lemma that $$A_rf(x)=\frac1{m(B(r,x))}\int_{B(r,x)}f(y)\,dy,$$where ##m## is Lebesgue measure, is jointly continuous in ##r## and ##x## (##A## stands for average). The claim that ##\chi_{B(r,x)}\to\chi_{B(r_0,x_0)}## on ##\mathbb R^n\setminus S(r_0,x_0)## is made in the...

A measure space ##(X,\mathcal M,\mu)## is complete iff $$S\subset N\in\mathcal M\text{ and }\mu(N)=0\implies S\in\mathcal M.$$The meaning of a complete measure is a measure whose domain includes all subsets of null sets. Suppose now ##\mu## is complete. Under what conditions is ##\nu## also...

Here's an excerpt from the proof of the change of variables formula in Folland's book (Theorem 2.47, page 76, 2nd edition, 6th and later printings): For reference, see Theorem 2.40 below. I don't understand how he is using Theorem 2.40 in the quoted passage. Which part of Theorem 2.40 is he...

This is an exercise from Folland's book. Here's my attempt at showing ##\mathcal M_1\otimes\mathcal M_2\otimes\mathcal M_3=(\mathcal M_1\otimes\mathcal M_2)\otimes\mathcal M_3##. ##\subset##: since every measurable rectangle ##A_1 \times A_2## belongs to ##\mathcal M_1 \otimes \mathcal M_2##...

Consider a set ##X## and family of sets ##\mathcal E\subset\mathcal P(X)##. Let ##\mathcal E_1=\mathcal{E}\cup\{E^c:E\in\mathcal E\}## and then for ##j>1## define ##\mathcal E_j## to be the collection of all sets that are countable unions of sets in ##\mathcal E_{j-1}## or complements of such...

Let ##m## be Lebesgue measure and ##\mathcal L## the Lebesgue ##\sigma##-algebra (the complete ##\sigma##-algebra that includes the Borel ##\sigma##-algebra). Consider, Folland starts off by saying that the collection of open intervals is invariant under translations and dilations, so the same...

Below is Proposition 1.15 in Folland at the beginning of the section of Borel measures on ##\mathbb R## (he is trying to construct a measure from ##F##). Here the algebra ##\mathcal{A}## is the finite disjoint union of h-intervals, where h-interval is a set of the form ##(a,b]##, ##(a,\infty)##...

I'm working the above exercise on the Lebesgue criterion for Riemann integrability in Folland's real analysis text, i.e. a function is Riemann integrable on ##[a,b]## iff it is continuous a.e. on ##[a,b]##. I think I know a solution, but I'm more concerned about elementary things. Are ##h,H##...

That's the first sentence in the proof. Prior to this Folland mentions the spaces ##L^1(\overline{\mu})## and ##L^1(\mu)## and how "we can (and shall) identify these spaces." (here ##\overline{\mu}## is the completion of ##\mu##). The propositions mentioned in the proof read as follows: I'm...

I'm reading in my probability book about characterizations of the law of a random variable, that is, the probability measure ##\mathbb P_X(A)=\mathbb P(X\in A)##. I read the following passage (I'm paraphrasing slightly): This extract is basically saying that if $$\mathbb E[\varphi(X)]:=\int...

Questions: 1. I am a bit unsure why ##g(x)=h(x)##. Clearly ##g(x)\geq h(x)##, but why is ##g(x)\leq h(x)##? Here's my explanation, which is kind of lengthy, but maybe you have a better one. If ##(a,b)\in\mathcal E_{\varphi}## is such that ##\varphi(x)>ax+b## for all ##x\in\mathbb R##, then...

I am reading a proof of Jensen's inequality. The proof goes like this. I do not know much about convex functions, but why does (1) hold? The definition of convex I'm using is that $$\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y)$$ holds for all ##x,y\in\mathbb R## and all ##t\in[0,1]##.

I'm working through an exercise to prove various identities of the conditional expectation. One of the identities I need to show is the following $$E(f(X,Y)\mid Y=y)=E(f(X,y)\mid Y=y).$$ But I am a little concerned about this identity from things I've read elsewhere. I am paraphrasing from...

Let ##(\Omega,\mathcal{F},P)## be a probability space, and let us define the conditional expectation ##{\rm E}[X\mid\mathcal{G}]## for integrable random variables ##X:\Omega\to\mathbb{R}##, i.e. ##X\in L^1(P)##, and sub-sigma-algebras ##\mathcal{G}\subseteq\mathcal{F}##. If...

I'm reading this article on transformation of random variables, i.e. functions of random variables. We have a probability space ##(\Omega, \mathcal F, P)## and measurable spaces ##(S, \mathcal S)## and ##(T, \mathcal T)##. We have a r.v. ##X:\Omega\to S## and a measurable map ##r:S\to T##. Then...

I'm reading in my book about the pdf of the sum of two continuous random variables ##X_1,X_2##. First, I'm a bit confused about the fact that the sum of two continuous random variables may not be continuous. Does this fact make the derivation below still valid or is there some key assumption...

Folland remarks on page 35 that each increasing and right-continuous function gives rise to not only a Borel measure ##\mu_F##, but also a complete measure ##\bar\mu_F## which includes the Borel ##\sigma##-algebra. He then says that the complete measure is the extension of the measure and that...

Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...

I wonder, how does one show that the integral is independent of the representation of the simple function? Suppose $$\phi=\sum_{i=1}^N c_i\chi_{E_i}=\sum_{i=1}^M b_i\chi_{F_i}.$$ How does it follow then that $$\sum_{i=1}^N c_i\mu(E_i)=\sum_{i=1}^M b_i\mu(F_i)?$$ I have discussed this problem...

I'm reading in these notes the following passage (I only have a question about the last two sentences): The last two sentences confuse me. Which sets does the author have in mind? I know what an algebra is (basically a sigma algebra, but not closed under countable infinite operations, but...

I am reading these notes on measure theory. On page 27, in chapter 2 on the construction of the Lebesgue measure, in section 2.8 on linear transformations, the author presents a lemma which is not proved. I wonder, how can one prove this? The author uses the following terminology; a (closed)...

Some definitions: The following statement has been left as an exercise in transfinite induction in a handout. I'm looking at Wikipedia and am trying to follow their outline: 1. Show it for the base case, i.e. that ##\mathcal{F}_{0}\subset\mathcal{G}##. This is, however, trivial, since we...

The notes I'm reading are from here. But I have summarized all the necessary details in this post. My question concerns the proposition, but it uses the definition below and the lemma. We say two rectangles are almost disjoint if they intersect at most along their boundaries. I omit the...

I was reading page 33 of https://staff.fnwi.uva.nl/p.j.c.spreij/onderwijs/TI/mtpTI.pdf when I saw this claim: Given measurable spaces ##(\Omega_1,\Sigma_1), (\Omega_2,\Sigma_2)## and the product space ##(\Omega_1\times \Omega_2, \Sigma)## where ##\Sigma## is the product sigma algebra, the...

my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}

I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...

Hi, I would like to know if an undergraduate student in physics could be able to study measure theory in order to have a better understanding of the probability theory and go further in this way (stochastic process) ? Assuming a first year of calculus and the level of "Mathematical methods in...

The claim states the following: Let ##(X,\mathcal{A},\mu)## be a measurable space, ##E## is a measurable subset of ##X## and ##f## is a measurable bounded function which has a bounded support in ##E##. Prove that: if ##f\ge 0## almost everywhere in ##E##, then for each measurable subset...

Hello everyone, My question for this thread concerns the application of (mainly) mathematical analysis to fields such as signal processing and machine learning. More specifically, I was wondering if you happen to know of some interesting application of things like measure theory or functional...

I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \begin{equation} \label{eq1} \begin{split} \frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...

I am an incoming graduate student in Theoretical Physics at Universiteit Utrecht, and I struggle to make a choice for one of my mathematical electives. I hope someone can help me out. My main interests lie in the fields of Statistical Physics, phase transitions and collective and critical...

I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...

Hi I am sitting with a homework problem which is to show if I can actually integrate a function. with 2D measure of lebesgue. the function is given by ##\frac{x-y}{(x+y)^2} d \lambda^2 (x,y)##. I know that a function ##f## is integrable if ##f \in L^{1}(\mu) \iff \int |f|^{1} d \mu < \infty##...

Problem: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if there is a $F_\sigma$ set $F \subset E$ with $m^*\left(F\right)=m^*\left(E\right)$. Proof: "$\leftarrow$" To Show: $E=K\cup N$ where $K$ is $F_\sigma$ and $m^*(N)=m(N)=0$. By assumption, $\exists F$, and...

I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A -> R be a continuous function. If f is non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...

Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that: Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...

##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...

Hi! Can anyone recommend a good introductory book for measure theory? I've found Terence Tao's online book to be a good start, but would I be asking too much if I wanted something even more introductory? Ultimately I'm working toward Ergodic theory (and probability theory along the way) with...

Hello. I have problem with this integral : \lim_{n \to \infty } \int_{\mathbb{R}^+} \left( 1+ \frac{x}{n} \right) \sin ^n \left( x \right) d\mu_1 where ## \mu_1## is Lebesgue measure.

Hi. Is there some kind of measure theory generalized to uncountable unions? Of course one needs to take care how to make sense of sums over an uncountable index set. I was thinking about following formulation of the additivity property of the "measure": $$\mu\left(\bigcup_{i\in\ I}...

Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...

I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...