  Let $X_0$ be a compact Riemannian manifold with boundary endowed with a
oriented, measured even dimensional foliation with purely transverse boundary.
Let $X$ be the manifold with cylinder attached and extended foliation. We prove
that the $L^2$--measured index of a Dirac type operator is well defined and the
following Atiyah Patodi Singer index formula is true $$ind_{L^2,\Lambda}(D^+) =
<\widehat{A}(X,\nabla)Ch(E/S),C_\Lambda> +
1/2[\eta_\Lambda(D^{\mathcal{F}_\partial}) - h^+_\Lambda + h^-_\Lambda].$$ Here
$\Lambda$ is a holonomy invariant transverse measure,
$\eta_{\Lambda}(D^{\mathcal{F}_{\partial}})$ is the Ramachandran eta invariant
\cite{Rama} of the leafwise boundary operator and the $\Lambda$--dimensions
$h^\pm_\Lambda$ 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.
