For $t > 0$ let $\mu_t$ be the measure on $\R^+$ with density
$$
x\mapsto\frac{e^{-x}x^{t-1}}{\G(t)}~.
$$
Show that the Laplace transform of $\mu_t$ is given by $\o_t(y)=(1+y)^{-t}$. 2. Conclude that $\mu_s*\mu_t=\mu_{s+t}$. The measure $\mu_t$ is called $\G$-distributions with parameter $t$. 3. Compute $\int x\,\mu_t(dx)$.
3. Since $\o^\prime(y)=\int-xe^{-xy}\,\mu_t(dx)$, we have: $\int x\,\mu_t(dx)=-\o^\prime(0)=t$.