Let $((0, \infty), \mathcal{M}_1, \mu_1)$ be measure space, where $\mathcal{M}_1$ be Lebesgue measurable sets in $(0, \infty)$ and $$\mu_1(E)=\int_{E} r^{d-1} dr$$ be a measure on $\mathcal{M_1}.$
Let unit sphere $S^{d-1}=\{x\in \mathbb R^d:|x|=1\}.$ Given $E\subset S^{d-1},$ we define $\tilde{E}=\{ x\in \mathbb R^d: x/|x|\in E, 0<|x|<1\}.$
We say $E\in \mathcal{M}_2$ exactly when $\tilde{E}$ is a Lebesgue measurable subset of $\mathbb R^d.$ We define $$\mu_2(E)=\sigma(E)=d\cdot m(\tilde{E})$$ where $m$ is a Lebesgue measure on $\mathbb R^d$
Then $(S^{d-1}, \mathcal{M}_2, \mu_2=\sigma)$ is a measure space.
My Question: How to verify following integration formula for polar coordinates: $$\int_{\mathbb R^d} f(x) dx= \int_{S^{d-1}} \left( \int_0^{\infty} f(r\gamma) r^{d-1} dr \right) d\sigma(\gamma)$$ when $f=\chi_{E}$ where $E$ is Lebesgue measurable set in $\mathbb R^d$
My attempt: Let $f=\chi_{E}.$ Then LHS= $m(E)$ To compute RHS, we take $E=E_1\times E_2$ where $E_1\in \mathcal{M}_1$ and $E_2\in \mathcal{M}_2$. Now I do not know: how to compute
$$\int_0^{\infty} \chi_{E_1\times E_2}(r\gamma) r^{d-1}dr$$