Editing Null set (section)

{{Short description|Measurable set whose measure is zero}}
{{For2|the set with no elements|Empty set|the set of zeros of a function|Zero set}}

[[File:Sierpinski triangle.svg|thumb|The [[Sierpiński triangle]] is an example of a null set of points in <math>\mathbb R^2</math>.]]

In [[mathematical analysis]], a '''null set''' is a [[Lebesgue measurable set]] of real numbers that has '''[[Lebesgue measure|measure]] zero'''.  This can be characterized as a set that can be [[Cover (topology)|covered]] by a [[countable]] union of [[Interval (mathematics)|interval]]s of arbitrarily small total length. 

The notion of null set should not be confused with the [[empty set]] as defined in [[set theory]]. Although the empty set has [[Lebesgue measure]] zero, there are also non-empty sets which are null.  For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.

More generally, on a given [[measure space]] <math>M = (X, \Sigma, \mu)</math> a null set is a set <math>S \in \Sigma</math> such that <math>\mu(S) = 0.</math>