Measure theory Definition and 133 Threads

Hi there! It looks like you are trying to prove that the second derivatives of the magnetic potential function ##\mathbf{A}## belong to the class ##C(\mathbb{R}^3)##. This is a great question and involves some advanced mathematical techniques. One approach you can take is to use the dominated...

Homework Statement Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c. Homework Equations m(E) is ofcourese referring to the outer measure of E The Attempt at a...

Hi I am trying to teach myself Measure Theory and I am using the book: Real Analysis by Stein and Skakarchi from Princeton. I want to check if my answers to the questions are correct, so I am asking: Does anyone have the answers to the questions in chapter 1 ?

I'm trying to read Brian Hall's book "Quantum Theory for Mathematicians". While (I think) I have a basic grasp of most of the prerequisites, I don't know any measure theory. According to the appendix, presumed knowledge includes "the basic notions of measure theory, including the concepts of...

I am working on a problem##^{(1)}## in Measure & Integration (chapter on Product Measures) like this: Suppose that ##f## is real-valued and integrable with respect to 2-dimensional Lebesgue measure on ##[0, 1]^2## and also ##\int_{0}^{a} \int_{0}^{b} f(x, y) dy dx = 0## for all ##a, b \in...

I am reviewing this http://deeplearning.cs.cmu.edu/pdfs/Cybenko.pdf on the approximating power of neural networks and I came across a definition that I could not quite understand. The definition reads: where $I_n$ is the n-dimensional unit hypercube and $M(I_n)$ is the space of finite...

I am think what is the structure of generated ##\sigma##-algebra. Let me make it specific. How to represent ##\sigma(\mathscr{A})##, where ##\mathscr{A}## is an algebra. Can I use the elements of ##\mathscr{A}## to represent the element in ##\sigma(\mathscr{A})##?

Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure...

I have this question on outer measure from Richard Bass' book, supposed to be an introductory but I am lost: Prove that ##\mu^*## is an outer measure, given a measure space ##(X, \mathcal A, \mu)## and define ##\mu^*(A) = \inf \{\mu(B) \mid A \subset B, B \in \mathcal A\}## for all subsets...

I am working on this problem on measure theory like this: Suppose ##X## is the set of real numbers, ##\mathcal B## is the Borel ##\sigma##-algebra, and ##m## and ##n## are two measures on ##(X, \mathcal B)## such that ##m((a, b))=n((a, b))< \infty## whenever ##−\infty<a<b<\infty##. Prove that...

I am working on a problem like this: Suppose ##\mathscr A_1 \subset \mathscr A_2 \subset \ldots## are sigma-algebras consisting of subsets of a set ##X##. Give an example that ##\bigcup_{i=1}^{\infty} \mathscr A_i## is not sigma-algebra. I was told to work along finite sigma-algebras on...

1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X? 2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set) 3. If 2 is true, can we...

Hey! :o What book would you recommend me to read about measure theory and especially the following: Measure and outer meansure, Borel sets, the outer Lebesgue measure. The Cantor set. Properties of Lebesgue measure (translation invariance, completeness, regularity, uniqueness). Steinhaus...

Hi, I'm trying to show that Givien a probability triplet (\theta,F,P) with G\in F a sub sigma algebra E(E(X|G))=E(X) Now I want to use E(I_hE(X|G))=E(I_hX) for every h\in G since that's pretty much all I've for the definition of conditional expected value. I know this property should use the...

If f=g a.e f and g are equal except at a measurable set with measure zero If two functions are not equal a.e what will then the negation be? Will there have to exist a set that is measurable, and f is not equal to g on this set, and this set has not measure 0? Or will the entire set...

Homework Statement Let (R,M,m) be Lebesgue measure space in R. Given E contained in R with m(E)>0 show that the set E-E defined by E-E:={x in R s.t. exists a, b in E with x= a-b } contains an interval centered at the origin Homework Equations try to prove by contradiction and use...

Let \lambda(A) denote the measure of A and let \lambda^{*}(A) denote the outer measure of A and let \lambda_{*}(A) denote the inner measure of A Okay so the question is as follows: Suppose that A \cup B is measurable and that \lambda(A \cup B) = \lambda^{*}(A) + \lambda^{*}(B) < \infty Prove...

Hello there. The stochastic calc book I'm going through ( and others I've seen ) uses the phrase "\mathscr{F}-measurable" random variable Y in the section on measure theory. What does this mean? I'm aware that \mathscr{F} is a \sigma-field over all possible values for the possible values of...

I have a question on sub-additivity. For sets ##E## and ##E_j##, the property states that if ##E=\bigcup_{j=0}^{\infty}E_j## then ##m^*(E) \leq \sum_{j=0}^{\infty}m^*(E_j)##, where ##m^*(x)## is the external measure of ##x##. Since ##E\subset \bigcup_{j=0}^{\infty}E_j##, by set...

So the above is the problem and my idea of how to approach it. This problem comes from the section on the Countable Additivity of Integration and the Continuity of Integration, but I was not sure how to incorporate those into the prove, if you even need them for the result. I had no idea what...

Author: Donald Cohn Title: Measure Theory Amazon Link: https://www.amazon.com/dp/0817630031/?tag=pfamazon01-20

Hi all, I have a question about measure theory: Suppose we have probability space (\mathbb{R}^d,\mathcal{B}^d,\mu) where \mathcal{B}^d is Borel sigma algebra. Suppose we have a function u:\mathbb{R}^d\times \Theta\rightarrow \mathbb{R} where \Theta\subset\mathbb{R}^l,l<\infty and u is...

Hi, I was wondering whether if ∫f×g dμ=∫h×g dμ for all integrable functions g implies that f = h? Thanks

This is a simple question. On pages 5-6 of Measure Theory,Vol 1, Vladimir Bogachev he writes that: for E=(A\cap S)\cup (B\cap (X-S)) Now, he writes that: X-E = ((X-A)\cap S) \cup ((X-B)\cap (X-S)) But I don't get this expression, I get another term of ((X-B)\cap (X-A)) i.e, X-E =(...

Do you think having Bogachev's Measure Theory (vol. I) as a first exposure to measure theory sounds a good idea? I mean while I can understand well the concepts presented in the book, I find some techniques used in the proof section quite hard to follow. :confused:

I have always struggled in understanding probability theory, but since coming across the measure theoretic approach it seems so much simpler to grasp. I want to verify I have a couple basic things.So say we have a set χ. Together with a σ-algebra κ on χ, we can call (χ,κ) a measurable space...

Hello everyone, I needed to know more about measure theory so I'm reading in Rudin's Principle's of Mathematical Analysis, somewhere in the chapter, he says: We let E denote the family of all elementary subsets of $R^p$... E is a ring, but not a $\sigma$-ring. According to my understanding of...

Hi all, I am not sure that if I have posted this thread on right place but as the subject is related to the stochastic & measure theory therefore I am posting it here. Well, my question is that in the subject "Preferences, Optimal Portfolio Choice, and Equilibrium" the tutor has used the...

Hi all, I am reading Probability and Measure by Patrick Billingsley, and I am stuck at one example, please help me understanding it. http://desmond.imageshack.us/Himg201/scaled.php?server=201&filename=30935274.jpg&res=landing Ω=(0,1] My question is that how come the A^c = (0,a_1]U(a'_1...

Homework Statement My question is would I be allowed to say, if lf+-\phil<ε/(2\mu(E) then ∫E lf+-\phil<ε/2 Homework Equations E is the set in which we are integrating over. \mu is the measure \varphi is a simple function f+ is the non-negative part of the function f. The Attempt...

I am starting graduate school in applied math in the fall and am trying to decide if measure theory is necessary or important in terms of applied math and if so, what ares of applied math? I have taken two basic real analysis courses through multiple integrals, etc. and would like to focus on...

Homework Statement I was wondering if we Let E be some set such that f-1(E) is measurable then so is f-1(E)c.Homework Equations If the set A is measurable then so is its compliment. The Attempt at a Solution I think the statement is true because f-1(E) is just a set and thus its compliment...

question 1: if f is a countably additive set function (probability measure) defined on σ-algebra A of subsets of S, then which of the probability space "(f, A, S) is called events? question 2: define what we mean by algebra and σ-algebra? for this question in the second part do we have to...

RThe "canonical representation phi" (measure theory) (Royden) Homework Statement I need some help understanding the canonical representation of phi as described on p. 77 of Rodyen's 3rd edition. I've transcribed it below for those of you who don't own the book. Homework Equations The...

Homework Statement Let (X,\mathcal{B},\mu) be a measure space and g be a nonnegative measurable function on X. Set \nu (E)=\int_{E}g\,d\mu. Prove that \nu is a measure and \int f\, d \nu =\int fg\,d\mu for all nonnegative measurable functions f on X. The Attempt at a Solution I'm basically at...

Can the concepts of measure theory or probability theory be derived from logic in a complete fashion? Or, are the concepts of measure theory merely proven by arguments whose forms are logical? I'm looking to gain a complete understanding of measure theory, and I wonder if that means I have to...

Homework Statement I am looking for an example of a series of funtions: \sum g_n on \Re such that: \int_{1}^{2}\displaystyle\sum_{n=1}^{\infty}g_n(x) \, dx \neq \displaystyle\sum_{n=1}^{\infty} \, \int_{1}^{2} \, g_n(x) \, dx "dx" is the Lebesque measure. 2. The attempt at a solution I...

The canonical example of a function that is not Riemann integrable is the function f: [0,1] to R, such that f(x)=1 if x is rational and f(x)=0 if x is irrational ( i know some texts put this the other way around, but bear with me because i can reference at least one text that does not). Hence...

Hi. I have a proof of a very basic measure theory theorem related to the definition of a measure, and would like to ask posters if the proof is wrong. Theorem: If E is measurable, then \overline{E} is measurable and conversely. My Proof: Let's try the converse version first. m(E)=m(E \cap...

Guys, I'm taking real analysis starting with open, close, compact sets, and neighborhoods. Now I'm addict to rely on these concepts to do my proofs. In the future I will have to take Measure Theory. Can anyone give me a percentage indication for how many percent theorems are proven by the set...

Can anyone recommend a book(s) that covers these topics: Measure theory / lebesgue integration Hilbert Spaces Distributions PDE's The only material I have is the lecture notes and they are quite difficult to work through. I need to get the basics I think, before I will...

If i have a measurable set with positive measure, how do I prove that there are 2 elements who's difference is in Q~{0} (aka a rational number that isn't 0.

Homework Statement Let \mathcal{A} be σ-algebra over a set X, and μ a measure in \mathcal{A}. Let A_{n} \in \mathcal{A} with \sum_{n=1}^{\inf} \mu(A_{n})< \inf Show that this implies μ ({x \in X : x \in A_n for infinitely many n}) = 0 . The Attempt at a Solution I don't even see how is the...

hi, i am learing about measure theory and i am looking fore some good reference of the subjects ..

I saw this problem on this site a while back and started to think about it. I can't find the post so I'll start it anew. The problem is: can you have two disjoint sets dense on an interval so that the measure of each set on any interval of that interval is equal? That is, say you have A, B in...

Homework Statement Suppose f\in L^2[0,1] and \int_0^1f(x)x^n=0 for every n=0,1,2... Show that f = 0 almost everywhere. Homework Equations My friend hinted that he used the fact that continuous functions are dense in L^2[0,1], but I'm still stuck. The Attempt at a Solution I need...

Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given. Given: Suppose \mu is a finite Borel measure on R, and define f(x)=\int\frac{d\mu(y)}{\sqrt{\left|x-y\right|}} Prove f(x) is finite almost everywhere If I integrate I...

let f_n be series of borel functions. Explain why set B = {x: \sum_n f_n(x) is not convergent} is borel set. Proof, that if\int_R |F_n|dY \leq 1/n^2 for every n then Y(B) = 0.Y is lebesgue measure.for first part i thought that set of A={x: convergent} is borel, and B=X\A so it's also borel...

Homework Statement The question is from Stein, "Analysis 2", Chapter 1, Problem 5: Suppose E is measurable with m(E) < ∞, and E = E1 ∪ E2, E1 ∩ E2 = ∅. Prove: a) If m(E) = m∗(E1) + m∗(E2), then E1 and E2 are measurable. b) In particular, if E ⊂ Q, where Q is a finite cube, then...

Homework Statement Here's an old qualifying exam problem I'm a little stumped on: Let (X,\mu) be a \sigma-finite measure space and suppose f is a \mu-measurable function on X. For t > 0, let \[ \phi(t) = \mu(\{x \in X : |f(x)| < t \}). \] Prove that \[ \int_0^{\infty}...