The Atiyah-Patodi-Singer index formula for measured foliations

Let X0 be a compact Riemannian manifold with boundary endowed with an oriented, measured even dimensional foliation with purely transverse boundary. Let X be the manifold with cylinder attached and extended foliation. We prove that the L2-measured index of a Dirac type operator is well defined and the following Atiyah-Patodi-Singer index formula is trueindL2,Λ(D+)=〈Aˆ(X,∇)Ch(E/S),CΛ〉+1/2[ηΛ(DF∂)−hΛ++hΛ−]. Here Λ is a holonomy invariant transverse measure, ηΛ(DF∂) is the Ramachandran eta invariant (M. Ramachandran, 1993) [23] of the leafwise boundary operator and the Λ-dimensions hΛ± of the space of the limiting values of extended solutions is suitably defined using square integrable representations of the equivalence relation of the foliation with values on weighted Sobolev spaces on the leaves.