On the density theorems of Borel and Furstenberg

Groups of automorphisms of Lie groups: density properties, bounded orbits, and homogeneous spaces of finite volume

“…The following extends a result of S. P. Wang [21], see also [20]; 6.3 below is also an analog of results in [13]. Here the representations are unitary and infinite dimensional, while those of [13] were nonunitary and finite dimensional.…”

“…By 2.3, it is infixed. But by [13] (or in case (i), by [2]) these conditions imply that μ is G-fixed; therefore μ is a G-invariant measure.…”

“…One of these uses a recent generalization of the Borel density theorem, given in [13]. If G operates on X then it also operates on M e (X) = finite Borel measures with compact support, giving rise to an infinite dimensional linear representation.…”

“…Here the representations are unitary and infinite dimensional, while those of [13] were nonunitary and finite dimensional. and p h T = Tp h for all heH, then by the above we have ρ g T = Tp g for all gsG, as required.…”

An abstract Borel density theorem

“…The Borel density theorem [1] states that if G is a semisimple linear algebraic group/R and H is a discrete, or more generally a Euclidean closed subgroup such that G/H has finite volume (or more generally has property S) then the algebraic (Zariski) hull //* of H equals G. In [4] I proved certain generalizations of the Borel density theorem in various forms. Principally this was done for minimally almost periodic groups (Furstenberg's case [3]), complex analytic linear groups (done independently by a different method by S. P. Wang [5]) and real analytic linear groups G with radical R and with the property that G/R has no compact factors, R acts with real eigenvalues and H is a lattice in G. While there was a certain underlying unity to these results the methods seemed, on the surface, to be ad hoc.…”