The Atiyah-Patodi-Singer signature formula for measured foliations

Given a compact manifold with boundary X-0 endowed with a foliation F-0 transverse to the boundary, and which admits a holonomy invariant transverse measure Lambda, we define three types of signature for the pair ( foliation, boundary foliation): the analytic signature, denoted by sigma(Lambda,an) (X-0 , partial derivative X-0), is the leafwise L-2-Lambda-index of the signature operator on the extended manifold X obtained by attaching cylindrical ends to the boundary; the Hodge signature sigma(Lambda),Hodge (X-0 , partial derivative X-0) is defined using the natural representation of the Borel groupoid R of X on the field of square integrable harmonic forms on the leaves; and the de Rham signature, sigma(Lambda,dR) (X-0 , partial derivative X-0), defined using the natural representation of the Borel groupoid R-0 of X-0 on the field of the L-2-relative de Rham spaces of the leaves. We prove that these three signatures coincide sigma(Lambda,an) (X-0 , partial derivative X-0) = sigma(Lambda),Hodge (X-0 , partial derivative X-0) = sigma(Lambda,dR) (X-0 , partial derivative X-0) As a consequence of the index formula we proved in [Bull. Sci. Math. (2010), DOI 10.1016/j. bulsci.2010.10.003], we finally obtain the Atiyah-Patodi-Singer signature formula for measured foliations: sigma(Lambda,dR) (X-0 , partial derivative X-0) = < L(TF0), C-Lambda > + 1/2[eta(Lambda)(D-F0)]