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 ${\displaystyle \mu (S)=0.}$

Every finite or countably infinite subset of the real numbers ⁠${\displaystyle \mathbb {R} }$⁠ is a null set. For example, the set of natural numbers ⁠${\displaystyle \mathbb {N} }$⁠, the set of rational numbers ⁠${\displaystyle \mathbb {Q} }$⁠ and the set of algebraic numbers ⁠${\displaystyle \mathbb {A} }$⁠ are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose ternary expansion can be written using only 0s and 2s (see Cantor's diagonal argument), and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and iteratively removing a third of the previous set, thereby multiplying the length by 2/3 with every step.

The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.

A subset ${\displaystyle N}$ of the real line ${\displaystyle \mathbb {R} }$ has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in ${\displaystyle \mathbb {R} }$ if and only if:

(In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of ${\displaystyle A}$ for which the limit of the lengths of the covers is zero.)