Editing Measure (mathematics) (section)

==Definition==
[[File:Countable additivity of a measure.svg|thumb|300px|Countable additivity of a measure <math>\mu</math>: The measure of a countable disjoint union is the same as the sum of all measures of each subset.]]

Let <math>X</math> be a set and <math>\Sigma</math> a [[σ-algebra]] over <math>X.</math> A [[set function]] <math>\mu</math> from <math>\Sigma</math> to the [[extended real number line]] is called a '''measure''' if the following conditions hold:

*'''Non-negativity''': For all <math>E \in \Sigma, \ \ \mu(E) \geq 0.</math>
*<math>\mu(\varnothing) = 0.</math>
*'''Countable additivity''' (or [[sigma additivity|σ-additivity]]): For all [[countable]] collections <math>\{ E_k \}_{k=1}^\infty</math> of pairwise [[disjoint sets]] in Σ,<math display="block">\mu{\left(\bigcup_{k=1}^\infty E_k\right)} = \sum_{k=1}^\infty \mu(E_k).</math>

If at least one set <math>E</math> has finite measure, then the requirement <math>\mu(\varnothing) = 0</math> is met automatically due to countable additivity:
<math display=block>\mu(E)=\mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing),</math>
and therefore <math>\mu(\varnothing)=0.</math>

If the condition of non-negativity is dropped, and <math>\mu</math> takes on at most one of the values of <math>\pm \infty,</math> then <math>\mu</math> is called a ''[[signed measure]]''.

The pair <math>(X, \Sigma)</math> is called a ''[[measurable space]]'', and the members of <math>\Sigma</math> are called '''measurable sets'''.

A [[tuple|triple]] <math>(X, \Sigma, \mu)</math> is called a ''[[measure space]]''. A [[probability measure]] is a measure with total measure one – that is, <math>\mu(X) = 1.</math> A [[probability space]] is a measure space with a probability measure.

For measure spaces that are also [[topological space]]s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in [[analysis (mathematics)|analysis]] (and in many cases also in [[probability theory]]) are [[Radon measure]]s. Radon measures have an alternative definition in terms of linear functionals on the [[locally convex topological vector space]] of [[continuous function]]s with [[Support (mathematics)#Compact support|compact support]]. This approach is taken by [[Nicolas Bourbaki|Bourbaki]] (2004) and a number of other sources. For more details, see the article on [[Radon measure]]s.