  Many index theorems (both classical and in noncommutative geometry) can be
interpreted in terms of a Lie groupoid acting properly on a manifold and
leaving an elliptic family of pseudodifferential operators invariant. Alain
Connes in his book raised the question of an index theorem in this general
context. In this paper, an analytic index for many such situations is
constructed. The approach is inspired by the classical families theorem of
Atiyah and Singer, and the proof generalizes, to the case of proper Lie
groupoid actions, some of the results proved for proper locally compact group
actions by N. C. Phillips.
