analysis - Property of Laplace transforms

I was looking at this answer to the question asked and I am curious about the

$$\int_0^\infty F(u)g(u) du = \int_0^\infty f(u)G(u) du$$

relationship being used. I referred to the link provided in the answer, but it discussed the Mellin transform, not the Laplace transform, and I am not making the connection.

I was wondering if someone could explain where this relationship comes from or where I could learn more about it, and if it is true for laplace transforms in general, or only for that specific problem.