  We define what it means for a proper continuous morphism between groupoids to
be Haar system preserving, and show that such a morphism induces (via pullback)
a *-morphism between the corresponding convolution algebras. We proceed to
provide a plethora of examples of Haar system preserving morphisms and discuss
connections to noncommutative CW-complexes and interval algebras. We prove that
an inverse system of groupoids with Haar system preserving bonding maps has a
limit, and that we get a corresponding direct system of groupoid
$C^*$-algebras. An explicit construction of an inverse system of groupoids is
used to approximate a $\sigma$-compact groupoid $G$ by second countable
groupoids; if $G$ is equipped with a Haar system and 2-cocycle then so are the
approximation groupoids, and the maps in the inverse system are Haar system
preserving. As an application of this construction, we show how to easily
extend the Maximal Equivalence Theorem of Jean Renault to $\sigma$-compact
groupoids.
