  It is well known that real measures on the circle are characterized by their
Herglotz transform, an analytic function in the unit disc. Invariance of the
measure under N-multiplication translates into a functional equation for the
Herglotz transform. Using elements from the theory of Hardy spaces one gets a
somewhat surprising condition for a sequence of complex numbers to be the
Fourier coefficients of an N-invariant measure. Next, starting from any
atomless measure on the circle we construct atomless premesures of bounded
kappa-variation in the sense of Korenblum which are invariant under s given
pairwise prime integers. The relevant function kappa is a generalized entropy
function depending on s. The proof uses Korenblum's generalized Nevanlinna
theory. Passing to "kappa-singular measures" and extending these to elements in
a Grothendieck group of possibly unbounded measures on the circle, one obtains
generalized invariant measures which are carried by "kappa-Carleson" sets. The
range of this construction depends on interesting questions about cyclicty in
growth algebras of analytic functions on the unit disc. We also describe some
very formal relations with Witt vectors. For example the Artin-Hasse
p-exponential "is" a p-invariant premeasure of bounded kappa_1 variation.
