Integrals for braided Hopf algebras

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radfordʹs formula for the fourth power of the antipode is obtained. The relationship of integration with cross-product and transmutation is studied. The results apply to topological Hopf algebras which do not have an additive structure, e.g. a torus with a hole.