Distance measure of a function

Question

In Mattila's Fourier Analysis and Hausdorff Dimension, we have on page 59 that given a finite measure $\mu$ compactly supported on a set $A \subset \mathbb{R}^n$, we can define a finite measure $\delta(\mu)$ compactly supported on its distance set $D(A):=\left\{\lVert x-y \rVert : x,y \in A \right\}\subset \mathbb{R}$ as $$ \delta(\mu)(B)= \int \mu(\{y:|x-y|\in B\}) d \mu(x). $$ It is then claimed that we have, for any $f \in C_0^\infty$, $$ \delta{(f)}(r)= \int (\sigma_{r}^{n-1} *f)f, $$ where $\sigma_{r}^{n-1}$ is the typical measure on the $(n-1)$-dimensional sphere of radius $r$. How is $\delta(f)$ defined here? I thought we could only input a measure.