On the dimension of the space of integrals on coalgebras

We study the injective envelopes of the simple right C-comodules, and their duals, where C is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if C is right co-Frobenius, then the dimension of the space of left M-integrals on C is for any left C-comodule M of finite support, and the dimension of the space of right N-integrals on C is for any right C-comodule N of finite support. Some examples of integrals are computed for incidence coalgebras.