Function having a given Mellin transform

Summary

A function $f$ on $(0, \infty)$ has Mellin transform $m$ at a point $s$ when the integral $\int_0^\infty x^{s-1} f(x) \, dx$ converges absolutely and its value is $m$. The notion bundles the Mellin convergence of $f$ at $s$ with the value of the Mellin transform. It is convenient in proofs, because a formula such as $\mathcal{M}f(s) = \Gamma(s)$ then carries its convergence claim with it.