  Let $(X,d)$ be a compact metric space, and let an iterated function system
(IFS) be given on $X$, i.e., a finite set of continuous maps $\sigma_{i}$: $
X\to X$, $i=0,1,..., N-1$. The maps $\sigma_{i}$ transform the measures $\mu $
on $X$ into new measures $\mu_{i}$. If the diameter of $ \sigma_{i_{1}}\circ
>... \circ \sigma_{i_{k}}(X)$ tends to zero as $ k\to \infty $, and if
$p_{i}>0$ satisfies $\sum_{i}p_{i}=1$, then it is known that there is a unique
Borel probability measure $\mu $ on $X$ such that $\mu =\sum_{i}p_{i} \mu_{i}
\tag{*}$.
  In this paper, we consider the case when the $p_{i}$s are replaced with a
certain system of sequilinear functionals. This allows us to study the variable
coefficient case of (*), and moreover to understand the analog of (*) which is
needed in the theory of wavelets.
