Proposition 16

Let A be a set equipped with a filter $\mathscr{F}$. For every $\alpha \in A$, let $\mu_\alpha$ be a bounded positive measure on $\mathbf{R}_+$, and let $\Phi_\alpha$ be the function defined on $\mathbf{R}_+$ by $\Phi_\alpha(p) = \int_0^{+\infty} e^{-px} d\mu_\alpha(x)$. In order that the mapping $\alpha \mapsto \mu_\alpha$ converge tightly with respect to $\mathscr{F}$ to a bounded positive measure $\mu$, it is necessary and sufficient that $\Phi_\alpha$ converge pointwise on $\mathbf{R}_+$, with respect to $\mathscr{F}$, to a function $\Phi$ continuous at the point 0. In this case, $\Phi(p) = \int_0^{+\infty} e^{-px} d\mu(x)$ for all $p \in \mathbf{R}_+$.

There are analogous results for the additive monoids $\mathbf{R}_+^n$ ($n$ an integer $> 1$); we leave to the reader the transcription of Th. 3 in this case.