  The left and right homological integrals are introduced for a large class of
infinite dimensional Hopf algebras. Using the homological integrals we prove a
version of Maschke's theorem for infinite dimensional Hopf algebras. The
generalization of Maschke's theorem and homological integrals are the keys to
study noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.
