# Measurable representations of semi simple Lie groups

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?

This is true and due to Béla von Szőkefalvi-Nagy, Über meßbare Darstellungen Liescher Gruppen (1936). 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).