IV - Transverse Measures

Summary

In this chapter we concentrate upon the measure theoretic aspects of foliated spaces, including especially the notion of transverse measures.

We begin with a general study of groupoids, first in the measurable and later in the topological context. Our examples come from the holonomy groupoid of a foliated space (2.20) and a discrete version corresponding to a complete transversal. We introduce transverse measures ν with a given modulus and discuss when these are invariant.

Next we look in the tangential direction, defining a tangential measure λ to be a collection of measures λ = { λ^x} (one for each leaf in the case of a foliated space) which satisfies certain invariance and smoothness conditions. For instance, a tangential, tangentially elliptic operator D yields a tangential measure ^lD as follows. Restrict D to a leaf l. It follows from Chapter I that KerD` and Ker D^*_lare locally finite-dimensional and hence the local index.

Next we specialize to topological groupoids and continuous Radon tangential measures. In the case of a foliated space we recount the Ruelle–Sullivan construction of a current associated to a transverse measure and we show that the current is a cycle if and only if the transverse measure is invariant.