Let $\mu$ be a Borel measure on $\mathbb R^n$ with compact support, and $0 < \mu(\mathbb R^n) < \infty$. We define the lower derivative and derivative of $\mu$ at $x\in \Bbb R^n$ by $$\underline{D}(\mu,x) = \liminf_{r\to 0} \frac{\mu(B(x,r))}{\alpha(n)r^n}$$ and $$D(\mu,x) = \lim_{r\to 0} \frac{\mu(B(x,r))}{\alpha(n)r^n}$$ respectively, the latter if it exists. Here, $\alpha(n)$ is the volume of the $n$-dimensional unit ball in $\mathbb R^n$.
What is the motivation for defining derivatives of measures in the above manner, and how does the above definition relate to the usual notion of the derivative in $\mathbb R^n$? Is there a way to recover the "original" notion of a derivative, from the above definition? I ask since this seems very out of the blue, and not well motivated.
Reference: Fourier Analysis and Hausdorff Dimension by Pertti Mattila.