  A general framework for the theory of statistical solutions on trajectory
spaces is constructed for a wide range of equations involving incompressible
viscous flows. This framework is constructed with a general Hausdorff
topological space as the phase space of the system, and with the corresponding
set of trajectories belonging to the space of continuous paths in that phase
space. A trajectory statistical solution is a Borel probability measure defined
on the space of continuous paths and carried by a certain subset which is
interpreted, in the applications, as the set of solutions of a given problem.
The main hypotheses for the existence of a trajectory statistical solution
concern the topology of that subset of "solutions", along with conditions that
characterize those solutions within a certain larger subset (a condition
related to the assumption of strong continuity at the origin for the Leray-Hopf
weak solutions in the case of the Navier-Stokes and related equations). The aim
here is to raise the current theory of statistical solutions to an abstract
level that applies to other evolution equations with properties similar to
those of the three-dimensional Navier-Stokes equations. The applicability of
the theory is illustrated with the B\'enard problem of convection in fluids.
