What shown below is a reference from "Analysis on manifolds" by James R. Munkres
Definition
Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if for every $\epsilon >0$ there is a covering $Q_1,Q_2,...$ of $A$ by countably many rectangles such that $$ \sum_{i=1}^\infty v(Q_i)<\epsilon $$ so that if this inequality holds, we often say that the total volume of the rectangles $Q_1,Q_2,...$ if less than $\epsilon$
Lemma 1
A set $A$ has measure zero in $\Bbb{R}^n$ if and only if for every $\epsilon>0$ there is a countable covering of $A$ by open rectangles $\overset{°}Q_1,\overset{°}Q_2,...$ such that $$ \sum_{i=1}^\infty v(Q_i)<\epsilon $$
Lemma 2
Let $Q$ be a rectangle in $\Bbb{R}^n$; let $\{Q_1,...,Q_k\}$ be a finite collection of rectangles that covers $Q$. Then $$ v(Q)\le\sum_{i=1}^k v(Q_i) $$
Statement
For any rectangle $R:=[a_1,b_1]\times...\times[a_n,b_n]$ the following things are equivalent:
- $R$ has measure zero;
- the volume of $R$ is zero;
- the interior of $R$ is empty.
Unfortunately I can't prove the statement so I ask to do it. So could someone help me, please?