The problem reads: Let F be the closed interval obtained by removing a countable collection of disjoint open intervals from a closed interval [a,b], where the sum of the lengths of such intervals is b-a. Show that F is of measure zero
Hint: F is covered by the finite collection of intervals obtained by removing a finite number of thos e disjoint intervals from [a,b].
So far, the book has only introduced the notion of Size in R(b-a for any interval [a,b] closed or open) and the definition of Zero Measure set both the classical way(always finding a covering s.t. that covering has size as small has we want) and the equivalent using step function integrals.
A set Z has measure zero in B if for any ε>0 we can find a nondecreasing sequence of step functions with integral(in B) each <ε and for all x in Z exists some order p s.t. n>p implies that h_n(x)>=1(step function).
I can´t seem to find any way to start cracking at this, and i have the feeling it´s pretty obvious.
Any hint would be highly appreciated!