1. ## Measure theory

Let $F:R^+ \to R$ given by F(x) = x + floor(x), show that for $f: R^+ \to R$ that ( $\mu_F$ Lebesgue-Stieltjes measure, that is $\mu_F(]a,b]) = F(b) - F(a)$)

$\int f d\mu_F = \int_0^\infty f(x) dx$ (= the normal Lebesgue integral) $+ \sum_{n=1}^\infty f(n)$

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?

2. I would prove this first for the characteristic function of an interval, then for simple functions (finite linear combinations of characteristic functions of intervals). Every integrable function is the pointwise limit of an increasing sequence of simple functions, so you can use the monotone convergence theorem for Lebesgue–Stieltjes integration to deduce the result in general.