Editing Null set (section)

==Measure-theoretic properties==

Let <math>(X,\Sigma,\mu)</math> be a [[measure space]]. We have:

* <math>\mu(\varnothing) = 0</math> (by [[Measure_(mathematics)#Definition|definition]] of <math>\mu</math>).
* Any [[countable]] [[union (set theory)|union]] of null sets is itself a null set (by [[Measure_(mathematics)#Countable_subadditivity|countable subadditivity]] of <math>\mu</math>).
* Any (measurable) subset of a null set is itself a null set (by [[Measure_(mathematics)#Monotonicity|monotonicity]] of <math>\mu</math>).

Together, these facts show that the null sets of <math>(X,\Sigma,\mu)</math> form a [[Sigma-ideal|𝜎-ideal]] of the [[sigma-algebra|𝜎-algebra]] <math>\Sigma</math>. Accordingly, null sets may be interpreted as [[negligible set]]s, yielding a measure-theoretic notion of "[[almost everywhere]]".