The following is an exercise from Bruckner's Real Analysis:
Let $F$ be the Cantor function. Show that, for every Borel set $E$, $μ_F(E) = \int_E F' dλ + μ_F(E∩K)$, where $K$ is the Cantor ternary set.
I have an answer in mind for this but it looks too simple and non-rigorous to be true! :
$\int_E F' dλ = \int_{E \setminus K} F' dλ = 0$ because $λ(K)=0$ and $F'=0$ on $E \setminus K$. $μ_F(E) = μ_F(E \setminus K) + μ_F(E∩K) = 0 + μ_F(E∩K)$ then just replace $0$ by $\int_E F' dλ$ because it is zero too. Am I right?