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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?

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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).

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Yes, see the Encyclopaedia of Math article: http://www.encyclopediaofmath.org/index.php/Finite-dimensional_representation

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  • $\begingroup$ This asserts the fact, but does not seem to give a reference (perhaps it is implied that this is in [7], but I can't even find this in MathSciNet [maybe because of a Russian-English transliteration issue]). Can you give me a reference, preferably in English, French, or German? $\endgroup$ – Sean Dec 29 '14 at 4:59
  • $\begingroup$ Well, you asked if it was true, I gave you an answer with a credible reference. It would be correct to upvote. $\endgroup$ – Igor Rivin Dec 29 '14 at 19:18
  • $\begingroup$ I don't have enough reputation to up vote. Sorry. $\endgroup$ – Sean Dec 30 '14 at 0:11
  • $\begingroup$ The link no longer works $\endgroup$ – Yemon Choi Oct 13 '17 at 1:59

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