Null set

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, a null set is a <a href="/facts/Lebesgue_measurable_set/8us9cGoW">Lebesgue measurable set</a> of real numbers that has <a href="/facts/Lebesgue_measure/8us9cGoW">measure</a> zero. This can be characterized as a set that can be <a href="/facts/Cover_(topology)/2VUdelA4">covered</a> by a <a href="/facts/Countable/Wj4DmWPv">countable</a> union of <a href="/facts/Interval_(mathematics)/e9se3vTe">intervals</a> of arbitrarily small total length. 
The notion of null set should not be confused with the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> as defined in <a href="/facts/Set_theory/1ptfsvUP">set theory</a>. Although the empty set has <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> 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 <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> 
 
 
 
 M
 =
 (
 X
 ,
 Σ
 ,
 μ
 )
 
 
 {\displaystyle M=(X,\Sigma ,\mu )}
 
 a null set is a set 
 
 
 
 S
 ∈
 Σ
 
 
 {\displaystyle S\in \Sigma }
 
 such that 
 
 
 
 μ
 (
 S
 )
 =
 0.
 
 
 {\displaystyle \mu (S)=0.}

Null set open-in-new

Null set