Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in


I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple
Lie groups, Annals of Mathematics 52 (1950), 509–517.


that measurable unitary representations of $G$ are actually continuous. Is this also true for finite-dimensional non-unitary representations?



Measurable representations of semi simple Lie groups
This is true and due to Béla von Szőkefalvi-Nagy, [*Über meßbare Darstellungen Liescher Gruppen* (1936)](http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002278286). Generalized to finite-dimensional representations of locally compact groups in A. Weil, *L'intégration dans les groupes topologiques* (1940, p. 66). Also exposed in Hewitt-Ross, *Abstract harmonic analysis* (1963, p. 346) or Fell-Doran, [*Representations of $\*$-algebras, etc.* (1988, p. 236)](https://books.google.com/books?id=tf_JCe5YPq8C&pg=PA236).


