Definition:Null Set

Definition
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

A set $N \in \mathcal A$ is called a ($\mu$-)null set iff $\mu \left({N}\right) = 0$.

Family of Null Sets
The family of $\mu$-null sets, $\left\{{N \in \mathcal A: \mu \left({N}\right) = 0}\right\}$, is denoted $\mathcal{N}_{\mu}$.

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 $\{I_i\}_{i \geq 1}$ of N-dimensional intervals such that:
 * $\displaystyle E \subseteq \bigcup_{i = 1}^\infty I_i$

and
 * $\displaystyle \sum_{i = 1}^\infty \operatorname{vol} \left({I_i}\right) \leq \epsilon$.

Here, $\operatorname{vol} \left({I}\right)$ denotes the volume of the interval $I_i$, 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 $N$-dimensional intervals 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.

Note
Not to be confused with the empty set.