Editing Measure (mathematics) (section)

===Sigma-finite measures===
{{Main|Sigma-finite measure}}

A measure space <math>(X, \Sigma, \mu)</math> is called finite if <math>\mu(X)</math> is a finite real number (rather than <math>\infty</math>). Nonzero finite measures are analogous to [[probability measure]]s in the sense that any finite measure <math>\mu</math> is proportional to the probability measure <math>\frac{1}{\mu(X)}\mu.</math> A measure <math>\mu</math> is called ''σ-finite'' if <math>X</math> can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a ''σ-finite measure'' if it is a countable union of sets with finite measure.

For example, the [[real number]]s with the standard [[Lebesgue measure]] are σ-finite but not finite. Consider the [[closed interval]]s <math>[k, k+1]</math> for all [[integer]]s <math>k;</math> there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the [[real number]]s with the [[counting measure]], which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the [[Lindelöf space|Lindelöf property]] of topological spaces.{{OR inline|date=May 2022}} They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.