Editing Measure (mathematics) (section)

{{short description|Generalization of mass, length, area and volume}}
{{For|the coalgebraic concept|Measuring coalgebra}}
{{Distinguish|Metric (mathematics)}}
{{More footnotes|date=January 2021}}
[[File:Measure illustration (Vector).svg|alt=|thumb|Informally, a measure has the property of being [[Monotone function|monotone]] in the sense that if <math>A</math> is a [[subset]] of <math>B,</math> the measure of <math>A</math> is less than or equal to the measure of <math>B.</math> Furthermore, the measure of the [[empty set]] is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.]]

In [[mathematics]], the concept of a '''measure''' is a generalization and formalization of [[geometrical measures]] ([[length]], [[area]], [[volume]]) and other common notions, such as [[magnitude (mathematics)|magnitude]], [[mass]], and [[probability]] of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in [[probability theory]], [[integral|integration theory]], and can be generalized to assume [[signed measure|negative values]], as with [[electrical charge]]. Far-reaching generalizations (such as [[spectral measure]]s and [[projection-valued measure]]s) of measure are widely used in [[quantum physics]] and physics in general.

The intuition behind this concept dates back to [[ancient Greece]], when [[Archimedes]] tried to calculate the [[area of a circle]].<ref>Archimedes [https://web.archive.org/web/20040703122928/http://www.math.ubc.ca/~cass/archimedes/circle.html Measuring the Circle]</ref><ref>{{Cite book |last=Heath |first=T. L. |url=http://archive.org/details/worksofarchimede029517mbp |title=The Works Of Archimedes |date=1897 |publisher=Cambridge University Press. |others=Osmania University, Digital Library Of India |pages=91–98 |chapter=Measurement of a Circle}}</ref> But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of [[Émile Borel]], [[Henri Lebesgue]], [[Nikolai Luzin]], [[Johann Radon]], [[Constantin Carathéodory]], and [[Maurice Fréchet]], among others.