Null set

In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals 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 ${\displaystyle M=(X,\Sigma ,\mu )}$ a null set is a set ${\displaystyle S\in \Sigma }$ such that ${\displaystyle \mu (S)=0.}$