Well I am working something, which deals with the following problem: For example, I want to compute an integral $\int\_{B(0,R)}f(x)dx$, where $B(0,R)=\{x\in\mathbb R^n:\;|x|\leq R\}$ and $S(0,R)=\{|x|=R\}$. Now we have the following formula

$$
\int\_{B(0,R)}f(x)dx=\int\_0^Rdr\int\_{S(0,r)}f(y)dS(y)
$$

My question is: I assume only that $f(x)$ is a measurable respect to Lebesgue measure and non-negative. Does the above formula hold? How do understand exactly the integral $\int\_{S(0,r)}f(y)dS(y)$? Could I consider $\int\_{S(0,r)}f(y)dS(y)$ as the Lebesgue integral of a Lebesgue measurable function $f$ on a Lebesgue measurable $S(0,r)$?