  A general theory of summation of divergent series based on the
Hardy-Kolmogorov axioms is developed. A class of functional series is
investigated by means of ergodic theory. The results are formulated in terms of
solvability of some cohomological equations, all solutions to which are
nonmeasurable. In particular, this realizes a construction of a nonmeasurable
function as first conjectured by Kolmogorov.
