I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}\_0(S)$, the space of projective classes of measured geodesic laminations with compact support on the surface $S$.
This space is compact (it is stated for example in the article "On Teichmueller’s metric and Thurston’s asymmetric metric on Teichmueller space" by Papadopoulos and Theret) but I can not find a proof of this fact. 


I've checked the uncompleted book "Closed curves on surfaces" by Bonahon: there it is proven that $\mathcal{PML}(S)$ (projective classes of measured geodesic laminations) is compact, but then I should prove that $\mathcal{PML}\_0(S)$ is closed in $\mathcal{PML}(S)$ and I have no clues. So:


**Is $\mathcal{PML}\_0(S)$ closed in $\mathcal{PML}(S)$? Why?**


Also, the proof of the compactness of $\mathcal{PML}(S)$ is given in a metric independent way (considering geodesic laminations as points in the boundary of the universal cover) which is less natural to me than the metric dependent fashion.


**Can you provide me of a reference of a proof of the compactness of $\mathcal{PML}\_0(S)$ in a metric dependent way?**



Why is $\mathcal{PML}_0(S)$ compact?
The original proof is given in Theorem 8.10.5 of ["The Geometry and Topology of 3-manifolds"](http://library.msri.org/books/gt3m/) by W. P. Thurston.


