Definition:Null Set

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

A set $N \in \Sigma$ is called a ($\mu$-)null set $\map \mu N = 0$.

Family of Null Sets
The family of $\mu$-null sets, $\set {N \in \Sigma: \map \mu N = 0}$, is denoted $\NN_\mu$.

Definition in $\R^n$
A set $E \subseteq \R^n$ is called a null set if for any $\epsilon > 0$ there exists a countable collection $J_i := \paren {\openint {\mathbf a_i} {\mathbf b_i} }$, $i \in \N$ of open $n$-rectangles such that:
 * $\ds E \subseteq \bigcup_{i \mathop = 1}^\infty J_i$

and
 * $\ds \sum_{i \mathop = 1}^\infty \map {\operatorname{vol} } {J_i} \le \epsilon$.

Here, $\map {\operatorname{vol} } {J_i}$ denotes the volume of the open rectangle $J_i$, which is the product of the lengths of its sides.

Said another way, a null set is a set that can be covered by a countable collection of open $n$-rectangles having total volume as small as we wish.

On Equivalence of Definitions of Null Set in Euclidean Space, it is shown that this definition is compatible with that for general measure spaces.

Also known as
Because of the defining equality $\map \mu N = 0$, a $\mu$-null set $N$ is also sometimes called a ($\mu$-)measure zero set.

Note
Not to be confused with the empty set.