 Solution

The convolution theorem for Laplace transforms stateswhen $x(t)={\int }_0^{t}y(s)z(t-s)ds$ and the transforms converge appropriately. One proves it by changing variables $t=s+r$ in the double integral over $0\le s\le t<\infty$, which becomes the product domain $s,r\ge 0$. Absolute convergence justifies the interchange.