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.

A null set is not to 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 ${\displaystyle \mu (S)=0.}$