Editing Support (mathematics) (section)

==In probability and measure theory==
{{details|Support (measure theory)}}

In [[probability theory]], the support of a [[probability distribution]] can be loosely thought of as the [[Closure (mathematics)|closure]] of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a [[sigma algebra]], rather than on a topological space.

More formally, if <math>X : \Omega \to \R</math> is a random variable on <math>(\Omega, \mathcal{F}, P)</math> then the support of <math>X</math> is the smallest closed set <math>R_X \subseteq \R</math> such that <math>P\left(X \in R_X\right) = 1.</math>

In practice however, the support of a [[discrete random variable]] <math>X</math> is often defined as the set <math>R_X = \{x \in \R : P(X = x) > 0 \}</math> and the support of a [[continuous random variable]] <math>X</math> is defined as the set <math>R_X = \{x \in \R : f_X(x) > 0 \}</math> where <math>f_X(x)</math> is a [[probability density function]] of <math>X</math> (the [[#set-theoretic support|set-theoretic support]]).<ref>{{cite web|last1=Taboga|first1=Marco|title=Support of a random variable|url=https://www.statlect.com/glossary/support-of-a-random-variable|website=statlect.com|access-date=29 November 2017}}</ref> 

Note that the word {{em|support}} can refer to the [[logarithm]] of the [[likelihood function|likelihood]] of a probability density function.<ref>{{cite book|first=A. W. F.|last=Edwards|title=Likelihood|edition=Expanded|location=Baltimore|publisher=Johns Hopkins University Press|year=1992|isbn=0-8018-4443-6|pages=31–34|url=https://books.google.com/books?id=LL08AAAAIAAJ&pg=PA31 }}</ref>