The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.
Riemann-type definitions of the Riemann improper integral and of the Lebesgue improper integral are obtained from McShane’s definition of the Lebesgue integral by imposing a Kurzweil-Henstock’s condition on McShane’s partitions.
The family Pλ of absolutely continuous probabilities w.r.t. the σ-finite measure λ is equipped with a structure of an infinite dimensional Riemannian manifold modeled on a real Hilbert. Firstly, the relation between the Hellinger distance and the Fischer metric is analysed on the positive cone Mλ+ of bounded measures absolutely continuous w.r.t. λ, appearing as a flat Riemannian manifold. Secondly, the statistical manifold Pλ is seen as a submanifold of Mλ+ and Amari-Chensov α-connections are derived. Some α-self-parallel curves are explicitely exhibited.