  We study statistical solutions of the incompressible Euler equations in two
dimensions with vorticity in $L^p$, $1\leq p \leq \infty$, and in the class of
vortex-sheets with a distinguished sign. Our notion of statistical solution is
based on the framework due to Bronzi, Mondaini and Rosa. Existence in this
setting is shown by approximation with discrete measures, concentrated on
deterministic solutions of the Euler equations. Additionally, we provide
arguments to show that the statistical solutions of the Euler equations may be
obtained in the inviscid limit of statistical solutions of the incompressible
Navier-Stokes equations. Uniqueness of trajectory statistical solutions is
shown in the Yudovich class.
