We consider solutions to the two-dimensional incompressible Euler system with
only integrable vorticity, thus with possibly locally infinite energy. With
such regularity, we use the recently developed theory of Lagrangian flows
associated to vector fields with gradient given by a singular integral in order
to define Lagrangian solutions, for which the vorticity is transported by the
flow. We prove strong stability of these solutions via strong convergence of
the flow, under only the assumption of L^1 weak convergence of the initial
vorticity. The existence of Lagrangian solutions to the Euler system follows
for arbitrary L^1 vorticity. Relations with previously known notions of
solutions are established
