We study two boundaries for the Teichmüller space of a surface Teich(S) due to L. Bers and W. Thurston. Each point in Bers's boundary is a hyperbolic 3-manifold with an associated geodesic lamination on S, its end-invariant, while each point in Thurston's is a measured geodesic lamination, up to scale. When dimℂ(Teich(S))>1, we show that the end-invariant is not a continuous map to Thurston's boundary modulo forgetting the measure with the quotient topology. We recover continuity by allowing as limits maximal measurable sublaminations of Hausdorff limits and enlargements thereof.
Jeffrey F. Brock. "Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds." Duke Math. J. 106 (3) 527 - 552, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10634-0