  We discuss various infinite-dimensional configuration spaces that carry
measures quasiinvariant under compactly-supported diffeomorphisms of a manifold
M corresponding to a physical space. Such measures allow the construction of
unitary representations of the diffeomorphism group, which are important to
nonrelativistic quantum statistical physics and to the quantum theory of
extended objects in d-dimensional Euclidean space. Special attention is given
to measurable structure and topology underlying measures on generalized
configuration spaces obtained from self-similar random processes (both for d =
1 and d > 1), which describe infinite point configurations having accumulation
points.
