  The Lagrangian average (LA) of the ideal fluid equations preserves their
fundamental transport structure. This transport structure is responsible for
the Kelvin circulation theorem of the LA flow and, hence, for its potential
vorticity convection and its helicity conservation.
  Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations.
