Lebesgue outer measure
Lebesgue outer measure is the concrete recipe for the smallest total length of open intervals needed to cover a set from outside. You try every possible way of blanketing the set with a countable collection of intervals, add up the lengths in each blanket, and take the infimum over all blankets. That infimum is the outer measure of the set.
Precisely, for a subset E of R, the Lebesgue outer measure is m-star(E) = inf of sum over n of length(I_n), where the infimum is taken over all countable collections of open intervals I_1, I_2, ... whose union contains E. This m-star is defined for every subset of R without exception, is monotone, and is countably subadditive — that is, it is a genuine outer measure in the abstract sense.
From this single formula the entire edifice of Lebesgue measure follows. Applying Carathéodory’s criterion to m-star produces the Lebesgue measurable sets, and m-star restricted to them is countably additive: it is Lebesgue measure. The honest subtlety is that m-star alone is only subadditive on arbitrary sets; full additivity is recovered only after restricting to the measurable ones.
To see m-star(Q intersect [0,1]) = 0: enumerate the rationals as r_1, r_2, ..., put an interval of length epsilon/2^n around r_n; the total length is epsilon, and epsilon is arbitrary, so the infimum is 0. The same trick shows every countable set has outer measure zero.
Cover a countable set by shrinking intervals of total length epsilon — for every epsilon.