  We consider the family known as modified or generalized surface
quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface
quasi-geostrophic (SQG) equation together with a family of regularized active
scalars given by introducing a smoothing operator of nonzero but possibly
arbitrarily small degree. This family naturally interpolates between the 2D
Euler equation and the SQG equation. For this family of equations we construct
an invariant measure on a rough $L^2$-based Sobolev space and establish the
existence of solutions of arbitrarily large lifespan for initial data in a set
of full measure in the rough Sobolev space.
