Measure theory Definition and 136 Threads

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}...

Hello all, I have a few questions in my mind: 1) \lim_{n\rightarrow \infty}[0,n) = \cup_{n\in\mathbb{N}}[0,n) = [0,infty) holds, and for \lim_{n\rightarrow \infty}[0,n] = \cup_{n\in\mathbb{N}}[0,n] = [0,infty) is also true? It should not be [0,infty] , am I correct? 2)...

Hey guys, below is a small question from introductory measure theory. Maybe be completely wrong on this, so if you could point me in the right direction I'd really appreciate it. Claim: Let B=\mathbb{Q} \cap [0,1] and \{I_k\}_{k=1}^n be a finite open cover for B. Then \sum_{k=1}^n m^*(I_k)...

Homework Statement Show that there is a continuous , strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero Homework Equations The Attempt at a Solution I need to find a mapping to a countable set or cantor set but I...

Homework Statement Let A and B be bounded sets for which there is \alpha > 0 such that |a -b| \geq\alpha for all a in A and b in B. Prove that outer measure of ( A \bigcup B ) = outer measure of (A) + outer measure of (B) Homework Equations We know that outer measure of the union is...

Hi All, I am a new phd student in engineering, working in signals analysis in neuroscience who seems to be doing a lot of work in statistics and probability theory. My uni is offering a course in measure theory. The course profile says: "The course is an introduction to measure theory and...

This is a practice final exam problem that has been giving me fits: Let E be a Lebesgue measurable subset of the interval [0,1] that has finite measure. Show that there exist two points x,y \in E such that x-y is rational.

Homework Statement Show that if E \subset B and B \in L(\mathbb{R}) (where L(R) denotes the family of Lebesgue measurable sets on the reals) with m(B) < \inf , then E \in L(\mathbb{R}) if and only if m(B) = m^{*}(E) + m^{*}(B - E), where m^* denotes the Lebesgue outer measure.Homework Equations...

One possible definition of measurability is this: A set E \subseteq \mathbb R^d is (Lebesgue) measurable if for every \epsilon > 0 there exists an open set \mathcal O \supseteq E such that m_*(\mathcal O \setminus E) < \epsilon. Here, m_* indicates Lebesgue outer measure. Apparently, an...

Hi everyone! my problem: since every simple function is bounded, we at once know, that either is our function f, cause: - \epsilon + g(x) <= f(x) <= \epsilon + g(x), so that's obviously not the problem here. this whole measure stuff doesn't get into my intuition and I don't have any...

Let (X, \mathcal{A}), (Y, \mathcal{B}) be measurable spaces. A function K: X \times \mathcal{B} \rightarrow [0, +\infty] is called a kernel from (X, \mathcal{A}) to (Y, \mathcal{B}) if i) for each x in X, the function B \mapsto K(x,B) is a measure on (Y, \mathcal{B}), and ii) for each B in...

Hello; Homework Statement Let \mathcal{A} be the \sigma-algebra on \mathbb{R}^2 that consists of all unions of (possibly empty) collections of vertical lines. Find the completion of \mathcal{A} under the point mass concentrated at (0,0). Homework Equations 1st: Completion is defined as...

All the books I want to read about probability and statistical estimation require some understanding of measure theory. What is a good introductory text on measure theory you would recommend (assuming no prior knowledge of measure theory at all)? I want to be able to teach myself from the book...

If you have two measurable sets A and B (not necessarily disjoint), is there an easy formula for the measure of the difference, m(A-B)?

Hi, I am pursuing graduate studies in economics, and I hear that "measure theory" is one of the classes that will impress admission commitees. I don't see anything by that name in my school's catalog. Does this class go by another name sometimes?

This question came up recently, and I'm wondering whether or not it's true: Let (X,A,m) be a finite measure space. Let E_1,E_2,... be a sequence of measurable subsets of X of constant positive measure (i.e., there exists c>0 such that m(E_i) = c for all i). Then there exists a subsequence of...

What theory are they? Set theory comes to mind but is that too broad?

Hi: I am trying to review the way L^p spaces are treated differently in Royden. In Ch.6, he treats them under "Classical Banach Spaces", and then again, in his Ch.11 , under "Abstract Spaces". This is what I understand: (Please comment/correct) In the case of abstract...

I was told that you can find a disjoint sequence of sets...say {Ei} such that m*(U Ei) < Σ m*(Ei).. That is the measure of the union of all these sets is less than the sum of the individual measure of each set... This is obvious if the sets aren't disjoint...But can someone give me an example...

Hi, I'm currently trying to teach myself some measure theory and I'm stuck on trying to show the following: Let (X,M,\mu) be a finite positive measure space such that \mu({x})>0 \forall x \in X . Set d(A,B) = \mu(A \Delta B), A,B \in X. Prove that d(A,B) \leq d(A,C) + d(C,B) . Could...

Problem: f_{n}\rightarrow f in measure, \mu(\left\{f_{n}>h\right\})\leq A Prove that \mu(\left\{f>h\right\})\leq A. My Work: Suppose not, then \mu(\left\{f>h\right\}) > A. From the triangle inequality for measures we get \mu(\left\{f>h\right\}) =...

How is measure theory associated with number theory, if at all. If they are connected, can anyone give a link?

How important is a measure theory course as an undergrad? I have to choose between taking an undergrad measure theory course and doing research. I'm already doing another research project, but I figure no grad school is going to penalize me for doing too much research. But how "bad" is it that...

if m(.) is a non-atomic measure on the Borel sigma-algebra B(I). I is some fixed closed finite interval in R. How to show that f satisfies the following: m(S) = L(f(S)), S in B(I) where L is the Lebesgue measure and f(x) = m( I intersect(-infinity,x] )

Homework Statement Let \mathscr{X} be a set, \mathscr{F} a \sigma-field of subsets of S, and \mu a probability measure on \mathscr{F}. Suppose that A_{1},\ldots,A_{n} are independent sets belonging to \mathscr{F}. Let \mathscr{F}_{k} be the smallest subfield of \mathscr{F} containing A_{1}...

Hi, Some time ago one of my professors told us about a remarkable theorem, which stated something along the lines of: if a function i takes two arguments, one being another function f, and the other being some region R on which the function f is defined, and this function i satisfies some...

Homework Statement Given is the measure space (A,\mathcal{P}(A),\mu) where \mu is the counting measure on the powerset \mathcal{P}(A) of A, i.e. \mu(E)=\#E I have to show that if \int_A f d\mu <\infty, then the set A_+=\{x\in A| f(x)>0\} is countable. 2. Relevant theorems I wish I knew...

1. The problem statement Let (X,M,\mu) be a measure space and let f:X \to [0,\infty] be a measurable function. Now define for E\in M the following function: \mu_f (E) = \int_E fd\mu Show that \mu_f is a measure on M. The Attempt at a Solution I will skip the part where I have to show that...

Hi, this is not a homework problem because as you can see, all schools are closed for the winter break. But I'm currently working on a problem and I'm not sure how to begin to attack it. Here's the entire problem: Let f be bounded and measurable function on [0,00). For x greater than or...

If E is a non empty set and (B_n)_{n \geq 1} are elements in the set 2^E. I then need help showing the following: lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcup_{n\, =\, 1} ^{\infty}\, B_n if and only if B_n\, \subseteq\, B_{n+1}, for all n\, \geq\, 1, Also I...

Definitions If X is a set, an algebra A on X is a non-empty collection of subsets of X which is closed under complements with respect to X, and finite unions. Given an algebra A, a premeasure on A is a function p\, :\, A \to [0,\, \infty] such that: a) p(\emptyset ) = 0 b) If B is a...

Sorry if this is kind of vague, but the other day, one of my math profs told me about a theorem which he thought was particularly interesting. I might be missing or getting a condition wrong, but here goes: Suppose I(f, d) is a real-valued function, where f is a real-valued function always...

If I have a sigma-algebra, A, consisting of subsets of X where X = {1,2,3,4}, and I also have a measure on A such that m({1,2}) = 1 m({1,2,3}) = 2 m({1,2,3,4}) = 3 Then my question is this: Is the set E = {3} a member of the sigma-algebra? I figured that since a subset E of X is in...

In a quantum gravity discussion ("Chunkymorphism" thread) some issues of basic topology and measure theory came up. Might be fun to have a thread for such discussions. for instance the statement was made, apparently concerning the real line (or perhaps more generally) that a countable set...