Equivalence of Definitions of Null Set in Euclidean Space

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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 $\left({J_i}\right)_{i \mathop \in \N}$ of $E$ by open $n$-rectangles, such that:
$\displaystyle \sum_{i \mathop = 1}^\infty \operatorname{vol} \left({J_i}\right) \le \epsilon$



Proof


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