Set of measure zero in ℝ^k

A set $N\subseteq\mathbb R^k$ has Lebesgue measure zero, or is a null set, if for every $\varepsilon>0$ there is a countable collection of $k$ -dimensional boxes $\{R_n\}_{n=1}^\infty$ such that

N \subseteq \bigcup_{n=1}^\infty R_n \quad\text{and}\quad \sum_{n=1}^\infty \operatorname{vol}(R_n) < \varepsilon.

For $R=\prod_{j=1}^k[a_j,b_j]$ , its volume is

\operatorname{vol}(R)=\prod_{j=1}^k(b_j-a_j).

(One may equivalently use open balls in place of rectangles; the definition is unchanged up to standard comparison arguments.)

Measure zero is a notion of smallness relevant to integration and differentiability. The Lebesgue criterion states that a bounded function on a compact rectangle is Riemann integrable if and only if its set of discontinuities has measure zero.

Examples

Any finite or countable subset of $\mathbb R^k$ has measure zero.
Any $k$ -dimensional affine hyperplane in $\mathbb R^{k+1}$ has measure zero in $\mathbb R^{k+1}$ .
The Cantor set has measure zero in $\mathbb R$ .