How can we calculate the following integral? $$ \int_{0}^r\frac{1}{s^n}\int_{B(s)}f(x)dxds $$ Here $B(s)$ is the ball of radius $s$ centered at the origin.
I think that this can be computed by $$ \int_{0}^r\frac{1}{s^n}\int_{B(s)}f(x)dxds =\int_{0}^r\frac{1}{s^n}\int_{0}^s\int_{\partial B(t)}f(x)dxdsdt $$ But I am stuck at this point. Any help is more than welcome.
You can apply Fubini at this step \begin{align*} \int_0^r s^{-n}\int_{B_s} f(x)\;dx\,ds &= \int_0^r s^{-n}\int_0^s \int_{\partial B_t} f(x)\;dS(x)\,dt\,ds\\ &= \int_0^r \int_t^r s^{-n}\int_{\partial B_t} f(x)\;dS(x)\, ds\, dt\\ &= \int_0^r \int_{\partial B_t} f(x)\; dS(x) \cdot \int_t^r s^{-n}\, dt\, ds\\ &= \int_0^r \int_{\partial B_t} f(x)\; dS(x) \cdot \frac{t^{1-n} - r^{1-n}}{n-1}\, ds\\ &= \int_0^r \int_{\partial B_t} \left(\frac{|x|^{1-n} - r^{1-n}}{n-1}\right)f(x)\; dS(x)\, dt\\ &= \int_{B_r} \left(\frac{|x|^{1-n} - r^{1-n}}{n-1}\right)f(x)\; dx\, dt\\ \end{align*}