Equivalence of Definitions of Null Set in Euclidean Space

Theorem
Let $\lambda^n$ be $n$-dimensional Lebesgue measure on $\R^n$.

Let $E \subseteq \R^n$.

Then the following are equivalent:


 * $(1):\quad \exists B \in \mathcal B \left({\R^n}\right): E \subseteq B, \lambda^n \left({B}\right) = 0$
 * $(2):\quad$ For every $\epsilon > 0$, there exists a countable cover $\left({J_i}\right)_{i \in \N}$ of $E$ by open $n$-rectangles, such that:
 * $\displaystyle \sum_{i = 1}^\infty \operatorname{vol} \left({J_i}\right) \leq \epsilon$