  We consider a complete metric space $(X,d)$ and a countable number of
contractive mappings on $X$, $\mathcal{F}=\{F_i:i\in\mathbb N\}$. We show the
existence of a {\em smallest} invariant set (with respect to inclusion) for
$\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\x) = r_i \boldmath{x} +
b_i$ on $X=\mathbb{R}^d$, we can prove a converse of the classic result on
contraction maps. Precisely, we can show that for that case, there exists a
{\em unique} bounded invariant set if and only if $r = \sup_i r_i$ is strictly
smaller than 1.
  Further, if $\rho = \{\rho_k\}_{k\in \mathbb N}$ is a probability sequence,
we show that if there exists an invariant measure for the system
$(\mathcal{F},\rho)$, then it's support must be precisely this smallest
invariant set. If in addition there exists any {\em bounded} invariant set,
this invariant measure is unique - even though there may be more than one
invariant set.
