Abstract. In this paper an abstract form of the Borel density theorem and related results is given centering around the notion of the author's of a (finite dimensional) "admissible " representation. A representation p is strongly admissible if each A'p is admissible. Although this notion is somewhat technical it is satisfied for certain pairs (G, p); e.g., if G is minimally almost periodic and p arbitrary, if G is complex analytic and p holomorphic. If G is real analytic with radical R, G/R has no compact factors and R acts under p with real eigenvalues, then p is strongly admissible. If in addition G is algebraic/R, then each R-rational representation is admissible. The results are proven in three stages where V is defined either over R or C. If p is a strongly admissible representation of G on V, then each (/-invariant measure fi on §(K), the Grassmann space of V, has support contained in the G-fixed point set. If p is a strongly admissible representation of G on V and G / H has finit
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