# What is Measure theory: Definition and 119 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.

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1. ### I On transformation of r.v.s. and sigma-finite measures

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...
2. ### I On pdf of a sum of two r.v.s and differentiating under the integral

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...
3. ### I On theorem 1.19 in Folland's and completion of measure

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...
4. ### I ##L^2## square integrable function Hilbert space

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...
5. ### I On integral of simple function and representation

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...
6. ### I Collection of finite unions of half-open intervals form an algebra

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...
7. ### I Parallel rectangles contained in oblique rectangle

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)...
8. ### I Help with basic transfinite induction proof

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...
9. ### I Two basic results from measure theory -- on volumes of rectangles

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...
10. ### I Prove projection of a measurable set from product space is measurable

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...
11. ### Approaching the Measure of a Set: Strategies for Finding f(Eα)

my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}
12. ### I Limit of limits of linear combinations of indicator functions

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...
13. ### Analysis Prerequisites Measure theory for ug student in physics

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...
14. ### I A claim in measure theory which seems flawed to me

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...
15. ### A Applications of analysis in signal processing/machine learning?

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...
16. ### I Radon-Nikodym Derivative and Bayes' Theorem

I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \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...
17. ### Courses Graduate level Mathematics courses of interest for Biological Physics

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...
18. ### A Definitions of Cylinder Sets and Cylinder Set Measure

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...
19. ### Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?

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##...
20. ### MHB Measure Theory - Existence of Fsigma set contained in measurable set

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...
21. ### I Munkres-Analysis on Manifolds: Extended Integrals

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...
22. ### I Munkres-Analysis on Manifolds: Theorem 20.1

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)...
23. ### A Convergence of a subsequence of a sum of iid r.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...
24. ### MHB Reference request - Measure theory

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...
25. ### A Convergence of an Integral Involving Lebesgue Measure and Sine Functions

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.

36. ### I Lebesgue measure and Fourier theory

Hi everyone, in this days i was seeing a little of Fourier series and transform, and i wondered if it was necessary to better understand before the measure and Lebesgue integral before studying it. Or it's not necessary?
37. ### Second derivatives of magnetic potential

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...
38. ### Proving Compact Set Exists with m(E)=c

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...
39. ### Analysis Answers to questions from the book: Real Analysis by Stein

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 ?
40. ### Analysis Basic measure theory for physics students

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...
41. ### Proving f = 0 almost everywhere

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...
42. ### What is meant when $\sigma$ is said to be discriminatory?

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...
43. ### Structure of generated sigma algbra

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})##?
44. ### Extension of measure on sigma-algebra

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...
45. ### How to prove the 2nd & 3rd conditions of outer 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...
46. ### How to prove a set belongs to Borel sigma-algebra?

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...
47. ### Union of increasing sigma-algebras is not sigma-algebra

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...
48. ### Is the Definition of Sigma Algebra Limited to Countable Unions?

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...
49. ### MHB Books to Learn Measure Theory Theory: Borel, Lebesgue, Cantor Set & More

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...
50. ### MHB Conditional expected value (using measure theory)

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...