Let given by F(x) = x + floor(x), show that for that ( Lebesgue-Stieltjes measure, that is )

(= the normal Lebesgue integral)

Any help is appreciated. How do I start? Do I start by figuring out how the measure looks for general sets? I can 'see' the result if the function is constant on an interval, but the normal representation can be very different from this, example: 1 on the cantor set and 0 everywhere else. How do I make sure those cases are included?