Integrating with respect to Poisson distribution

Question

If $F$ is a distribution with mean $\mu_1$ and with $F(0)=0$, how do I integrate $\displaystyle\int_0^\infty y\,F(\mathrm dy)\;$? This question is based on the compounded Poisson process, $S(t)=\sum^{N(t)}_{k=1}Y_k$, where $N(t)\sim\mathrm{Po}(\lambda_1)$and $Y_k$ are i.i.d. non-negative random variables with distribution $F(\cdot)$.