Why are there different methods of constructing the outer measure leading up to Lebesgue measure (open covering or closed covering?)?

Question

In studying outer measure (to build up to Lebesgue measure) I notice some authors construct the outer measure of $E$ as the infimum of the length/size of closed sets that cover $E$ (Royden) and sometimes as the infimum of the length/size of open sets that cover $E$ (Stein). Why the difference? Is one preferable over the other?

Answer 1

To add even more complication, Folland constructs Lebesgue measure with right-closed half-open intervals.

The distinction is irrelevant, as ultimately we want that the Lebesgue outer measure restricted to the Borel $\sigma$-algebra is the Lebesgue measure (and in fact, when restricted to an algebra of Borel sets, is the premeasure induced by the function $F(x) = x$ on that algebra).

Since the Borel sets in $\mathbb{R}$ can be generated by open sets, closed sets, left-closed h-intervals, or right-closed h-intervals, it really matters not what we choose.