I have seen the following identity, but I think that it is not correct:
Given a positive measure $\mu$ in $\mathbb{R}^N$, we have that $$\displaystyle\int_{\|x-y\|^{-\gamma}\leq R}\|x-y\|^{-\gamma}d\mu(y)=\displaystyle\int_0^R \mu\{y\in \mathbb{R}^N:\|x-y\|^{-\gamma}>t\} dt.$$
It seems to me that these must be an inequality. Could someone tell me if this is correct or not?
For any non-negative measurable function $f$ we have $\int f(y)d\mu (y)=\int_0^{\infty} \mu (\{y: f(y) >t\})dt$. This is a consequence of Fubini's Theorem. Just put $f(y)=1_{\|x-y\|^{-\gamma} \leq R} \|x-y\|^{-\gamma}$ in this equation and note that the integrand vanishes for $t >R$.
$\int_0^{\infty} \mu \{x: f (x) >t) dt=\int_0^{\infty} \int \chi_{\{f>t\}}(x) d\mu (x) dt=\int \int_0^{f(x)} dt d\mu(x)=\int f(x) d\mu(x)$. The interchange of integrals is justified by Tonelli's Theorem.