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:


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