Suppose a locally compact
group G (always second countable) has a Borel action on an analytic Borel
space S so that each element of G transforms a given measure μ into an
equivalent measure. If S0 is the coset space for a closed subgroup H, then
there is a natural action of G on S0 which comes from translations of G
on itself and there is such a quasi-invariant measure. Thus it is reasonable
to think of such a space (S,μ), for some purposes, as a generalized sort of
subgroup, or a virtual subgroup of G. In fact, the set S × G can be given
algebraic and measure-theoretic structure so that many of the procedures
used with subgroups can be carried over to this general setting. There is a
general notion of virtual group, not necessarily “contained in” a group, which
can be derived from this, and it turns out to include equivalence relations
with suitable measures as a special case. These virtual groups appear in
studying group representations, operator algebras, foliations, etc. Since there
is a general setting for virtual groups, it seems desirable to see whether
the intuitive idea of an action of a group as representing a subobject fits
into this framework in a compatible way. The purpose of this paper is to
show that “images” under homomorphisms, “kernels”, etc. do fit together
properly.
