In my text, the outer measure of a set $E \subseteq \mathbb{R}^d$ is defined using closed boxes or rectangles ${Q_k} = \prod [a_i, b_i]$ via:
$$ m^*(E) = \inf \left\{ \sum \text{vol}(Q_k); E \subseteq \cup_k Q_k \right\} $$
The problem is to show that the usual definition is equivalent to the above: instead of requiring that $E \subseteq \cup Q_k$, we could have $E \subseteq\cup Q_k^\circ$ just as well.
I know simply that that since $E \subseteq \cup_k Q_k^\circ\subseteq \cup_k{Q_k} $ we have $$ m^*(E) \leq \inf \left\{ \sum \text{vol}(Q_k); E \subseteq \cup_k Q_k^\circ \right\} $$ how to go about in the reverse direction?
Edit: My (incomplete) Solution Consider any cover $\{Q_k\}$ such that $E \subseteq \cup Q_k^\circ$. Take closed boxes $\{Q_k'\}$ in the interior of $\{Q_k\}$ such that $E \subseteq \cup Q_k' \subseteq \cup Q_k^\circ$ and $\text{vol}(Q_k') + \epsilon/2^k \geq \text{vol}(Q_k)$, then: $$ \sum_k \text{vol}(Q_k')\geq \sum_k\left(\text{vol}(Q_k)-\frac{\epsilon}{2^k}\right)=\sum_k \text{vol}(Q_k)-\epsilon \geq \inf \sum_k \text{vol}(Q_k)-\epsilon $$ at which point I am stuck.
You have the inclusion going the wrong way.
Define $$m_0^*(E) = \inf \left\{ \sum \text{vol}(Q_k); E \subseteq \cup_k Q_k^o \right\}.$$
If $E \subseteq \cup_k Q_k^o$ then (clearly) $E \subseteq \cup_k Q_k$ and thus $$m^*(E) \le \sum \text{vol}(Q_k).$$ Take the infimum over all coverings $\{Q_k^o\}$ to obtain $$m^*(E) \le m_o^*(E).$$
To prove the other inequality you may assume that $m^*(E) < \infty$ since otherwise there is nothing to show.
Let $\epsilon > 0$, let $E \subseteq \cup_k Q_k$ with $\sum \text{vol}(Q_k) < \infty$, and select rectangles $\{Q_k'\}$ with the property that $Q_k \subset Q_k'^o$ and $\text{vol}(Q_k') < \text{vol}(Q_k) + \epsilon/2^k$ for all $k$. Since $E \subseteq \cup_k Q_k'^o$ you get $$m_o^*(E) \le \sum \text{vol}(Q_k') < \sum \text{vol}(Q_k) + \epsilon.$$ Now take the infimum over all coverings $\{Q_k\}$ to obtain $$m_0^*(E) \le m^*(E) + \epsilon.$$ Finally let $\epsilon \to 0^+$ to get the desired inequality.