We extend to continuous groups
our recent results on strongly approximately transitive group actions. We are
concerned with locally compact second countable groups and standard Borel
G-spaces. A G-space 𝒳 with a σ-finite quasiinvariant measure α is called strongly
approximately transitive (SAT) if there exists a probability measure ν ≪ α such for
every Borel set A with α(A)≠0 and every 𝜀 > 0 one can find g ∈ G with
ν(gA) > 1 −𝜀. Examples of SAT G-spaces include boundaries of spread out random
walks on G. We prove that when G is compactly generated and has polynomial
growth then every standard SAT G-space (𝒳,α) is necessarily purely atomic;
when G is additionally connected, (𝒳,α) is a singleton. The Choquet-Deny
theorem for spread out random walks on G follows as a corollary. For a
connected G we establish the equivalence of the following conditions: (a) G has
polynomial growth; (b) every standard SAT G-space is a singleton; (c) every SAT
homogeneous space of G is a singleton; (d) every homogeneous space of G admits a
σ-finite invariant measure; (e) the Choquet-Deny theorem holds for every
spread out probability measure on G; (f) the Choquet-Deny theorem holds for
every absolutely continuous compactly supported probability measure on
G.
