I am seeking a “direct” proof that the Lebesgue measure of the irrationals on the unit interval is $1$. The standard proof I see is that the measure of the unit interval is 1, and the rationals have $0$ measure, so the irrationals must have measure $1$.
That is a perfectly good proof, but if we didn’t know the rationals have $0$ measure, is there a “direct” proof that the irrationals have measure $1$?