I am reading "Analysis on Manifolds" by James R. Munkres.
Definition:
In this book, we call $A$ has measure zero in $\mathbb{R}^n$ if for every $\epsilon > 0$, there is a countable covering of $A$ by closed rectangles $Q_1^{'}, Q_2^{'}, \cdots$ such that $$\sum_{i=1}^{\infty} v(Q_i^{'}) < \epsilon.$$
By the way, in this book, the author didn't write $[a_1, b_1] \times \cdots [a_n, b_n]$ is a closed rectangle or not if $a_i = b_i$ for some $i$. But I believe $[a_1, b_1] \times \cdots [a_n, b_n]$ is a closed rectangle when $a_i = b_i$ for some $i$.
There is the above Theorem 11.1(c) on p.91 in this book.
I think we need to modify the following part of the proof because if $v(Q_i^{'})$ is equal to $0$, then there is no $Q_i$ such that $Q_i^{'} \subset \text{Int } Q_i \text{ and } v(Q_i) \leq 2 v(Q_i^{'})$.
Cover $A$ by rectangles $Q_1^{'}, Q_2^{'}, \cdots$ of total volume less than $\frac{\epsilon}{2}$. For each $i$, choose a rectanble $Q_i$ such that $$Q_i^{'} \subset \text{Int } Q_i \text{ and } v(Q_i) \leq 2 v(Q_i^{'}).$$
I modified like the following:
Cover $A$ by rectangles $Q_1^{'}, Q_2^{'}, \cdots$ of total volume less than $\frac{\epsilon}{4}$. For each $i$, choose a rectanble $Q_i$ such that $$Q_i^{'} \subset \text{Int } Q_i \text{ and } v(Q_i) < \max \{2 v(Q_i^{'}), \frac{\epsilon}{2^{i+1}}\}.$$
If we modify like the above, then $$\sum v(Q_i) < \sum \max \{2 v(Q_i^{'}), \frac{\epsilon}{2^{i+1}}\} \leq \sum (2 v(Q_i^{'}) + \frac{\epsilon}{2^{i+1}}) = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$
I am not good at mathematics, so I am not sure I am right or not.
Am I right or not?