  A criterion for subnormality of unbounded composition operators in L2-spaces,
written in terms of measurable families of probability measures satisfying the
so-called consistency condition, is established. It becomes a new
characterization of subnormality in the case of bounded composition operators.
Pseudo-moments of a measurable family of probability measures that satisfies
the consistency condition are proved to be given by the Radon-Nikodym
derivatives which appear in Lambert's characterization of bounded composition
operators. A criterion for subnormality of composition operators induced by
matrices is provided. The question of subnormality of composition operators
over discrete measure spaces is studied. Two new classes of subnormal
composition operators over discrete measure spaces are introduced. A recent
criterion for subnormality of weighted shifts on directed trees by the present
authors is essentially improved in the case of rootless directed trees and
nonzero weights by dropping the assumption of density of C\infty-vectors in the
underlying L2-space.
