The Atiyah Patodi Singer signature formula for measured foliations

Let $(X_0,{\mathcal{F}}_0)$ be a compact manifold with boundary endowed with a foliation ${\mathcal{F}}_0$ which is assumed to be measured and transverse to the boundary. We denote by $\mathrm{\Lambda }$ a holonomy invariant transverse measure on $(X_0,{\mathcal{F}}_0)$ and by ${\mathcal{R}}_0$ the equivalence relation of the foliation. Let $(X,\mathcal{F})$ be the corresponding manifold with cylindrical end and extended foliation with equivalence relation $\mathcal{R}$. In the first part of this work we prove a formula for the $L^2$-$\mathrm{\Lambda }$ index of a longitudinal Dirac-type operator $D^{\mathcal{F}}$ on $X$ in the spirit of Alain Connes’ non commutative geometry $ind_{L^2,\Lambda}(D^{$\mathcal{F}$,+}) = <\hat{A}(T$\mathcal{F}$)Ch(E/S),C_\Lambda> + 1/2[\eta_\Lambda(D^{$\mathcal{F}$\partial}) - h^+{\Lambda} + h^-_{\Lambda}].$