From Mattila's Fourier Analysis and Hausdorff Dimension (page 59):
Theorem 4.6(a): If $A \subset \mathbb{R}^n$ is Borel and its dimension is greater than $(n+1)/2$, then the interior of $D(A):=\{\|x-y\|:x,y \in A\} \in \mathbb{R}$ is nonempty.
In the proof of this statement, Mattila introduces the concept of a distance measure $\delta(\mu)$ defined on $B \in \mathbb{R}$ induced by a $\mu \in \mathcal{M}(A)$ (the set of nonzero finite measures compactly supported on $A$) by defining
$$\delta(\mu):= \int\mu\{y: \|x-y\|\in B\} d\mu (x).$$
What is the idea behind this definition? I can't really wrap my mind around it. Also, how is it that we know that for any continuous $\varphi$ on $\mathbb{R}$, we have
$$\int \varphi d\delta(\mu)= \iint \varphi(\|x-y\|) d\mu(x)d\mu(y)?$$