Editing Probability measure (section)

==Definition==
[[File:Probability-measure.svg|thumb|300px|A ''probability measure'' mapping the σ-algebra for  <math>2^3</math> events to the [[unit interval]].]]
The requirements for a [[set function]] <math>\mu</math> to be a probability measure on a [[σ-algebra]] are that:

* <math>\mu</math> must return results in the [[unit interval]] <math>[0, 1],</math> returning <math>0</math> for the empty set and <math>1</math> for the entire space.
* <math>\mu</math> must satisfy the ''[[Sigma-additive set function|countable additivity]]'' property that for all [[countable]] collections <math>E_1, E_2, \ldots</math> of pairwise [[disjoint sets]]: <math display=block> \mu\left(\bigcup_{i \in \N} E_i\right) = \sum_{i \in \N} \mu(E_i).</math>

For example, given three elements 1, 2 and 3 with probabilities <math>1/4, 1/4</math> and <math>1/2,</math> the value assigned to <math>\{1, 3\}</math> is <math>1/4 + 1/2 = 3/4,</math> as in the diagram on the right.

The [[conditional probability]] based on the intersection of events defined as:
<math display=block>\mu (B \mid A) = \frac{\mu(A \cap B)}{\mu(A)}.</math>
satisfies the probability function requirements so long as <math>\mu(A)</math> is not zero.<ref>{{Cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://link.springer.com/book/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X}}</ref><ref>''Probability, Random Processes, and Ergodic Properties'' by Robert M. Gray 2009 {{isbn|1-4419-1089-1}} [https://books.google.com/books?id=x-VbL8mZWl8C&pg=PA163 page 163]</ref>

Probability measures are distinct from the more general notion of [[Fuzzy measure theory|fuzzy measures]] in which there is no requirement that the fuzzy values sum up to <math>1,</math> and the additive property is replaced by an order relation based on [[set inclusion]].