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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?

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The original proof is given in Theorem 8.10.5 of "The Geometry and Topology of 3-manifolds" by W. P. Thurston.

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