I have found excellent proof about changing variables in polar coordinateThis is the problemLet $S^{n-1}=\left\{x \in R^2:|x|=1\right\}$ and for any Borel set $E \in S^{n-1}$ set $E *=\{r \theta: 0<r<1, \theta \in E\}$. Define the measure $\sigma$ on $S^{n-1}$ by $\sigma(E)=n|E *|$.
With this definition the surface area $\omega_{n-1}$ of the sphere in $R^n$ satisfies $\omega_{n-1}=n \gamma_n=\frac{2 \pi^{n / 2}}{\Gamma(n / 2)}$, where $\gamma_n$ is the volume of the unit ball in $R^n$. Prove that for all non-negative Borel functions $f$ on $R^n$,$$\int_{R^n} f(x) d x=\int_0^{\infty} r^{n-1}\left(\int_{S^{n-1}} f(r \theta) d \sigma(\theta)\right) d r .$$
this is the solution, but I didn't catch what is underlined
Can someone prove it? Thanks
EDIT I was also proving another apprach. I thought that if $A$ is an open subset of $S_(n−1)$ if I call E be the set of all ru with $r_1 < r < r_2$ , $u \in A$, I verified that theformula holds for the characteristic function of E and now i wanted to pass from theseto characteristic functions of Borel sets in $\mathbb{R}^k$. but I don't know how to prove in detail... I only have the idea