  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 $\Lambda$ 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 $L^2$-$\Lambda$ index of a
longitudinal Dirac-type operator $D^{\mathcal{F}}$ on $X$ in the spirit of
Alain Connes' non commutative geometry
  $ind_{L^2,\Lambda}(D^{\mathcal{F},+}) =
<\hat{A}(T\mathcal{F})Ch(E/S),C_\Lambda> +
1/2[\eta_\Lambda(D^{\mathcal{F}_\partial}) - h^+_{\Lambda} + h^-_{\Lambda}].$
