Assuming that Polar Coordinates Formula (Theorem 2.49 of Folland Real Analysis) works for simples functions, how to obtain the result for $L^1$ functions? That is, suppose that there exists an measure on $S^{N-1}$ such that $$ \int_{\mathbb{R}^N} f(x) dx = \int_{0}^{+\infty}\int_{S^{N-1}} f(rx') r^{N-1} d\sigma dr, $$ for all SIMPLE function $f$. How to prove the same equality works for a Borel mensurable $f \in L^1$ ? I know how to conclude this for a Borel mensurable function $f \geq 0$. One just have to use approximation by increasing simple functions and Monotone Convergence Theorem. However I got stuck in the case $f \in L^1$, where we assume nothing about the positiveness of $f$.
Any help is welcome.