  We study the Euler-Poincar\'e equations that are the regularized Euler
equations derived from the Euler-Poincar\'e framework. It is noteworthy to
remark that the Euler-Poincar\'e equations are a generalization of two
well-known regularizations, the vortex blob method and the Euler-$\alpha$
equations. We show the global existence of a unique weak solution for the
two-dimensional (2D) Euler-Poincar\'e equations with the initial vorticity in
the space of Radon measure. This is a remarkable feature of these equations
since the existence of weak solutions with the Radon measure initial vorticity
has not been established in general for the 2D Euler equations. We also show
that weak solutions of the 2D Euler-Poincar\'e equations converge to those of
the 2D Euler equations in the limit of the regularization parameter when the
initial vorticity belongs to the space of integrable and bounded functions.
