This is a comment, not an answer, but I am not entitled. The short answer, if not directly to your question, is that this works fine if the limit set has the continuity property that its boundary has zero $\mu$ measure. This follows from early work of Grothendieck which can easily found with a search machine.

This is a comment, not an answer, but I am not entitled. The short answer, if not directly to your question, is that this works fine if the limit set has the continuity property that its boundary has zero $\mu$ measure. This follows from early work of Grothendieck which can easily found with a search machine.

Added on request: The wikipedia article „Weak convergence of measures“ gives the precise result and some references, though not specifically to Grothendieck. I am fairly sure that it is in his text on topological vector spaces but I don‘t have access to a copy. The classical text by Billingsley which is referenced in wiki is probably the best place to start.