  Let $X_0$ be a complete borderless infinite area hyperbolic surface. We
introduce Thurston's boundary to the Teichm\"uller space $T(X_0)$ of the
surface $X_0$ using Liouville (geodesic) currents. Thurston's boundary to
$T(X_0)$ is identified with the space $PML_{bdd}(X_0)$ of projective bounded
measured laminations on $X_0$ which naturally extends Thurston's result for
closed surfaces. Moreover, the quasiconformal mapping class group
$MCG_{qc}(X_0)$ acts continuously on the closure $T(X_0)\cup PML_{bdd}(X_0)$.
