  This is the fourth part in the series of articles math.MG/0503397,
math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is
developed. In this part it is shown that the filtration on valuations
introduced in math.MG/0503399 is compatible with the product. Then it is proved
that the Euler-Verdier involution on smooth valuations introduced in
math.MG/0503399 is an automorphism of the algebra of valuations. Then an
integration functional on valuations with compact support is introduced, and a
property of selfduality of valuations is proved. Next a space of generalized
valuations is defined, and some basic properties of it are proved. Finally a
canonical imbedding of the space of constructible functions on a real analytic
manifold into the space of generalized valuations is constructed, and various
structures on valuations are compared with known structures on constructible
functions.
