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.

Note
Not to be confused with the empty set.