Corollary

— Let $S$ be a full submonoid of $X$, such that the closure of $S \cap X_0$ contains 1. Let $L$ be the subset of $\mathscr{C}^b(S; \mathbf{C})$ consisting of the restrictions to $S$ of the Laplace transforms of the measures $\lambda \in \mathscr{M}_+^b(M)$.

a) The set $L$ is closed in the space $\mathscr{C}^b(S; \mathbf{C})$ equipped with the topology of pointwise convergence.

b) The mapping $\lambda \mapsto (\mathscr{L} \lambda)_S$ is a homeomorphism of $\mathscr{M}_+^b(M)$ onto $L$, if $\mathscr{M}_+^b(M)$ is equipped with the tight topology and $L$ with the topology of pointwise convergence.

c) The topology of pointwise convergence and the topology of compact convergence coincide on $L$.

The assertions a) and b) are immediate consequences of Th. 3; the assertion c) follows from b) and Prop. 14, since the topology of compact convergence is finer than that of pointwise convergence.

One must be on guard that L is not closed in the set of all bounded complex functions on S, equipped with the topology of pointwise convergence. Assume for example the notations of Example 2 below (M = \mathbf{R}_+, S identified with \mathbf{R}_+). The Laplace transforms of the measures $\varepsilon_n$ ($n \in \mathbf{N}$) are the functions $t \mapsto e^{-nt}$ on $\mathbf{R}_+$; as $n$ tends to $+\infty$, these functions converge pointwise to the function equal to 1 for $t = 0$ and to 0 for $t \neq 0$, which does not belong to L.

Example 1). — Take for M the set N of positive integers, equipped with the law of addition and with the discrete topology. Let D be the unit disc of \mathbf{C} (the set of complex numbers of absolute value $\leq 1$) equipped with the topology induced by \mathbf{C} and with the law induced by multiplication. For every $z \in D$, let us denote by $f(z)$ the character $n \mapsto z^n$ of N. For every character $\chi$ of N, denote by $g(\chi)$ the complex number $\chi(1) \in D$. One verifies immediately that $f$ and $g$ are mutually inverse homeomorphisms between D and X, which will permit us, from now on, to identify X and D. The set of characters tending to 0 at infinity may then be identified with the set $D_0$ of complex numbers of absolute value < 1. Finally, the interval ]0, 1] of \mathbf{R} is a full submonoid of D, and 1 is in the closure of $]0, 1[ \cap D_0 = ]0, 1[$.

Every measure $\mu$ on N may be written in a unique way in the form $\mu = \sum_{n \in \mathbf{N}} u_n \cdot \varepsilon^n$, and $\mu$ is bounded if and only if $\sum_n |u_n| < +\infty$; one then has $\mathscr{L}\mu(z) = \sum_{n \in \mathbf{N}} u_n z^n$ for $z \in D$. This function is continuous on D; it is customary to call it the generating function of the summable sequence $(u_n)_{n \in \mathbf{N}}$. Transcribed into this language, Th. 3 yields the following result (taking into account Prop. 9 of No. 3):