I am reading the Exterior Measure section by Stein and Shakarchi (2009).
In their definition an exterior of measure of any subset $E$ of $\mathbb{R}^d$ is
$$m_*(E)=\inf\sum^\infty_{j=1}|Q_j|.$$
where the infimum is taken over all countable coverings $E\subset\bigcup^\infty_{j=1}Q_j$.
My Question: I want to show $|Q|\leq m_*(Q) $. |.| is the volume of a cube. Why is it enough to show $|Q|\leq\sum^\infty_{j=1}|Q_j|$ for an aribtrary covering $Q\subset\bigcup_jQ_j$?
Isn't the following true, so where does sufficiency come from? What am I missing?
$$|Q|\leq\sum^\infty_j|Q_j|\leq\inf\sum^\infty_j|Q_j|.$$
Reference:
$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.
Best Answer
Expanding my comments into an answer:
Your last inequality is incorrect. It is not true that $$\sum^\infty_{j=1}|Q_j| \leq \inf\sum^\infty_{j=1}|Q_j|.$$ The opposite inequality is true: if we define $$\mathcal{C}_Q = \left\{\sum^\infty_{j=1}|Q_j|:(Q_j)_{j=1}^{\infty} \text{ a countable covering of $Q$}\right\},$$ then for any covering $(Q_j) \in \mathcal{C}_Q$ we have $$\inf\mathcal{C}_Q \leq \sum^\infty_{j=1}|Q_j|.$$
Once we have shown that $|Q| \leq \sum_{j=1}^{\infty}|Q_j|$ for any countable covering $(Q_j)$ in $\mathcal C_Q$, this will imply that $|Q|$ is a lower bound of $\mathcal{C}_Q$, and therefore $|Q|$ cannot be larger than the greatest lower bound of $\mathcal{C}_Q$, hence $$|Q| \leq \inf \mathcal{C}_Q \leq \sum_{j=1}^{\infty}|Q_j|.$$
Note that $\mathcal{C}_Q$ is a set of nonnegative numbers, not a set of sets. Each number in $\mathcal{C}_Q$ is the total volume $\sum|Q_j|$ of some countable covering of $Q$. Then $\inf \mathcal C_Q$ is the greatest lower bound of this set of numbers. It is not necessarily true that there is a covering $(Q_j)$ that achieves this lower bound. In general, for any covering $(Q_j)$ we will have strict inequality: $$\inf \mathcal C_Q < \sum |Q_j|$$ However, by definition of infimum / greatest lower bound, it is true that for any $\epsilon > 0$, we can find a covering $(Q_j)$ such that $$\inf \mathcal C_Q \leq \sum |Q_j| \leq \inf \mathcal C_Q + \epsilon$$