The Atiyah Patodi Singer Signature formula for measured foliations

Given a compact manifold with boundary $X_0$ endowed with a foliation $\mathcal{F}_0$ transverse to the boundary, and which admits a holonomy invariant transverse measure $\Lambda$, %Let $(X,\mathcal{F})$ be the corresponding manifold with % cylindrical ends and extended foliation with equivalence relation $\mathcal{R}$. In the paper \cite{io} we proved a formula for the $L^2$-$\Lambda$ index of a % longitudinal % Dirac-type operator $D^{\mathcal{F}}$ on $X$ in the spirit of Alain Connes' non commutative geometry \cite{Cos}. %Here $\Lambda$ is a holonomy invariant transverse measure, %$\eta_{\Lambda}(D^{\mathcal{F}_{\partial}}$ is the measured eta invariant of the boundary operator defined by Ramachandran \cite{???}, %and the $\Lambda$--dimension $h^{\pm}_{\Lambda}$ of the space of extended solution is defined using square integrable representations of the equivalence relation with values in the weighted $L^2$ spaces of the leaves. we define three types of signature for the pair (foliation, boundary foliation): the analytic signature, denoted $\sigma_{\Lambda,\operatorname{an}}(X_0,\partial X_0)$, is the leafwise $L^2$-$\Lambda$-index of the signature operator on the extended manifold $X$ obtained attaching cylindrical ends to the boundary; the Hodge signature \noindent $\sigma_{\Lambda,\operatorname{Hodge}}(X_0,\partial X_0)$ is defined using the natural representation of the Borel groupoid $\mathcal{R}$ of $X$ on the field of square integrable harmonic forms on the leaves; and the de Rham signature, $\sigma_{\Lambda,\operatorname{dR}}(X_0,\partial X_0)$, defined using the natural representation of the Borel groupoid $\mathcal{R}_0$ of $X_0$ on the field of the $L^2$ relative de Rham spaces of the leaves. We prove that these three signatures coincide $$\sigma_{\Lambda, \operatorname{an}}(X_0,\partial X_0)= \sigma_{\Lambda,\operatorname{Hodge}}(X_0,\partial X_0)=\sigma_{\Lambda,\operatorname{dR}}(X_0,\partial X_0).$$ As a consequence of the index formula we proved in \cite{io}, we finally obtain the main result of this work, the Atiyah-Patodi-Singer signature formula for measured foliations: $$\sigma_{\Lambda,\operatorname{dR}}(X_0,\partial X_0)=\langle L(T\mathcal{F}_0),C_{\Lambda}\rangle +1/2[\eta_{\Lambda}(D^{\mathcal{F}_{\partial}})].$$ We give also, in the appendix, an account of noncommutative integration theory.