Existence and Uniqueness of Orbital Measures

We note an elementary proof of the existence and uniqueness of a solution $$ to the equation $\mu =p\mu_{0}+q\hat{F}\mu$. Here $\mathbb{X}$ is a topological space, $\mathbb{P(X)}$ is the set of Borel measures of unit mass on $\mathbb{X}$, $\mu_{0}\in$ $\mathbb{P(X)}$ is given, $p>0$, and $q\geq 0$ with $p+q=1$. The transformation $\hat{F}:$ is defined by \hat{F}\upsilon =\tsum\limits_{n=1}^{N}p_{n}\upsilon \circ f_{n}^{-1} where f_{n}:\mathbb{% X\to X} is continuous, $p_{n}>0$ for $n=1,2,...,N$, $N$ is a finite strictly positive integer, and \tsum\limits_{n=1}^{N}p_{n}=1. This problem occurs in connection with iterated function systems (IFS).