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:

$(1):\quad \exists B \in \map \BB {\R^n}: E \subseteq B, \map {\lambda^n} B = 0$

$(2):\quad$ For every $\epsilon > 0$, there exists a countable cover $\family {J_i}_{i \mathop \in \N}$ of $E$ by open $n$-rectangles, such that:

$\ds \sum_{i \mathop = 1}^\infty \map {\operatorname {vol} } {J_i} \le \epsilon$

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Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 6$: Problem $8$