  We study statistical solutions of the incompressible Navier-Stokes equation
and their vanishing viscosity limit. We show that a formulation using
correlation measures, which are probability measures accounting for spatial
correlations, and moment equations is equivalent to statistical solutions in
the Foias-Prodi sense. Under the assumption of weak scaling, a weaker version
of Kolmogorov's self-similarity at small scales hypothesis that allows for
intermittency corrections, we show that the limit is a statistical solution of
the incompressible Euler equations. To pass to the limit, we derive a
Karman-Howarth-Monin relation for statistical solutions and combine it with the
weak scaling assumption and a compactness theorem for correlation measures.
