A note on translation continuity of probability measures (Q1184097)

Let \(G\) be a second countable locally compact group which acts continuously and transitively on a second countable locally compact space \(S\). Then \(S\) has a unique nontrivial invariant measure class \([\mu ]\). As the main result of this paper the author shows that a probability measure \(\nu\) on \(S\) is absolutely continuous with respect to some measure (or, equivalently, to all measures) contained in \([\mu ]\) if and only if \(\lim_{g\in G ,g\to e}\nu(gU) = \nu (U)\) for each open set \(U\) in \(S\). The case that \(G\) does not act transitively on \(S\) is also discussed.