Groups of quasi-invariance and the Pontryagin duality

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that (1) there exists a continuous monomorphism ϕ from G into X with dense image, and (2) for each g ∈ X either g ∈ ϕ ( G ) and the shift μ g is equivalent to μ or g ∉ ϕ ( G ) and μ g is orthogonal to μ. It is proved that ϕ ( G ) is a σ-compact subset of X. We show that there exists a Polish non-locally quasi-convex (and hence nonreflexive) QI-group such that its bidual is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of X. It is constructed a reflexive non-discrete group topology on the integers.