In Asmussen's Applied Probability and Queues, Proposition 2.7 makes a claim about the form of the renewal density in terms of the density of the interarrival distribution $F$, namely:
The renewal density $u$ exists if and only if $F$ has a density $f$. Then $u = \sum_{n=1}^{\infty} f^{\ast n}$ or, equivalently, $u$ is the solution of the renewal equation $u = f + F \ast u$.
The renewal density is originally defined as the density of the renewal measure $U = \sum_{i=1}^n F^{\ast n}$.
Asmussen's proof is one sentence long and leaves me completely in the dark. I'm not even sure where to start to manipulate these convolutions of measures to illuminate the proof. The first claim is easy to believe, but I'm not sure how one gets $u = U \ast f$ in the second claim. Can anyone provide a hint for where to start?
For each $t>0$, we have \begin{align} f(t)+F\star u(t) &= f(t) + \left[F\star\sum_{n=1}^\infty f^{\star n}\right](t)\\ &= f(t) + \sum_{n=2}^\infty f^{\star n}(t)\\ &= \sum_{n=1}^\infty f^{\star n}(t) = u(t), \end{align} from which it follows that $u$ satisfies the renewal equation $u=f+F\star u$.