What is Measure theory: Definition and 116 Discussions
In mathematics, a measure on a set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the usual length, area, or volume to subsets of a Euclidean spaces, for which this be defined. For instance, the Lebesgue measure of an interval of real numbers is its usual length.
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see § Definition, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.
In general, if one wants to associate a consistent size to all subsets of a given set, while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.
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...
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...
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...
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}...
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...
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...
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...