I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4).
Namely, why the Laplace Transform of the function: $$\int_{0}^{s}\psi(s-x)\cdot \overline{F}_{U}(x) \ dx$$ is equal to $$\mu \cdot \hat{L}_{\psi}(s)\cdot \hat{L}_{\overline{F_{U}^{s}}}(s)$$?
where:
$U$ is a nonnegative random variable with is a c.d.f. $F_{U}$ and expected value $\mu$
$\psi$ is a c.d.f. of ruin function,
$\overline{F}_{U}$ is a tail distribution of $U$, i.e. $\overline{F}_{U}=1-F_{U}$,
$F_{U}^{s}$ is defined as follows: $F_{U}^{s}(x)=\frac{1}{\mu}\int_{0}^{x}\overline{F}_{U}(y) \ dy$,
$\overline{F_{U}^{s}} = 1 - F_{U}^{s}$,
$\hat{L}_{\psi}$ is a notation for Laplace Transform of $\psi$, i.e. $\hat{L}_{\psi}(s)=\int_{0}^{\infty}\psi(y)e^{-sx}dx$,
$\hat{L}_{\overline{F_{U}^{s}}}$ - respectively
I noticed that this integral is a convolution.
Any help would be very appreciated. :)