Editing Mathematical analysis (section)

=== Measure theory ===
{{Main|Measure (mathematics)}}

A measure on a [[set (mathematics)|set]] is a systematic way to assign a number to each suitable [[subset]] of that set, intuitively interpreted as its size.<ref>{{cite book|author-link = Terence Tao|first = Terence|last = Tao|date = 2011|title = An Introduction to Measure Theory| series=Graduate Studies in Mathematics | volume=126 |publisher = American Mathematical Society|isbn = 978-0821869192|url = https://books.google.com/books?id=HoGDAwAAQBAJ|access-date = 2018-10-26|archive-date = 2019-12-27|archive-url = https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ|url-status = live|doi=10.1090/gsm/126}}</ref> 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 conventional [[length]], [[area]], and [[volume]] of [[Euclidean geometry]] to suitable subsets of the <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math>. For instance, the Lebesgue measure of the [[Interval (mathematics)|interval]] <math>\left[0, 1\right]</math> in the [[real line|real numbers]] is its length in the everyday sense of the word – specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or [[Extended real number line|+∞]] to (certain) subsets of a set <math>X</math>. It must assign 0 to the [[empty set]] and be ([[countably]]) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a ''consistent'' size to ''each'' subset 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 [[Sigma-algebra|<math>\sigma</math>-algebra]]. This means that the empty set, countable [[union (set theory)|unions]], countable [[intersection (set theory)|intersections]] and [[complement (set theory)|complements]] of measurable subsets are measurable. [[Non-measurable set]]s 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]].