Definition:Null Set

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 $$E \subseteq \bigcup_{i = 1}^\infty I_i$$ and $$\sum_{i = 1}^\infty \text{vol}(I_i) \leq \epsilon$$.

Here, $$\text{vol}(I)$$ 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.