In proving a Lemma regarding $m^{*}$, Tao ('An Introduction to Measure Theory') states that 'by definition of the Lebesgue outer measure we can cover $E\cup F$ by a countable family of boxes such that'
$$\sum_{n=1}^{\infty}m(B_{n})\leq m^{*}(E\cup F) +\epsilon$$
Question: What is the reason or proof of this result?
Since elementary sets are just finite unions of boxes, we can write
$$m^{*}(E\cup F)=\inf\left\{\sum_{n=1}^{\infty}m(B_{n}): B_{n}\text{ are boxes that cover $E\cup F$}\right\}$$
The way I intuitively explained it to myself was that since $m^{*}(E\cup F)$ is the infimum, we should be able to choose $B_{n}$'s to arbitrarily well approximate $m^{*}(E\cup F)$ from above so that
$$\sum_{n=1}^{\infty}m(B_{n}) – m^{*}(E\cup F)\leq \epsilon$$
Is this correct? And how would one more rigorously explain this? I know that for real numbers, if $a > b$, then there must exist $\epsilon>0$ so that $a>b+\epsilon$, but I am not sure if that is relevant here.
Best Answer
Your argument is rigorous enough. Maybe you have thought about it too hard.