Applications

This chapter defines equi-integrability, and states that any finite or convergent family is p -equi-integrable. It explains Vitali-s convergence theorem, and describes the converse to the dominated convergence theorem. The chapter proves the integrability and the convergence of the composite function of certain continuous mappings. It presents some properties of continuity with respect to increasing p . The chapter also characterizes the open subsets of a separable Banach space. It examines the measurability for values in such a space, and explains the Riesz theorem and composition of isomorphisms.