Abstract. It is shown that if $S$ is a commutative involution semigroup then the set ${\cal H}(S)$ of all moment functions on $S$ (i.e., complex-valued functions defined on $SS:=\{ st\mid s,t\in S \} $ and admitting a disintegration as an integral of hermitian multiplicative functions) is closed under pointwise convergence in ${\bf C}^{SS}$ if and only if for each $s$ in $S$ there is a positive integer $n$ such that $(s^*s)^n$ is the product of $2n+1$ elements of $S$. In fact, if the condition is satisfied then an `indeterminate method of moments' holds, asserting that if $(\varphi_i)$ is a net of moment functions, converging pointwise to some function $\varphi $, and if for each $i$ in the index set $\mu_i$ is a disintegrating measure of $\varphi_i$ then some subnet of $(\mu_i)$ converges to a disintegrating measure of $\varphi $ (implying in particular that $\varphi $ is a moment function).

AMS Subject Classification (1991): 43A35, 44A60

Received May 28, 2002, and in revised form December 12, 2002. (Registered under 2927/2009.)