Some remarks of topology and of measure theory applied to the definition of the Ito integral

Abstract

In the final part of the paper the concept of the Ito integral is revisited. The emphasis is not as much in the resulting definition, which still contains some variations with respect to the usual ones, as in the approach. The novelty of this latter has many facets. First the appropiate dense subspace of the domain space of the integral is obtained through a more sound and simple technique, after establishing a preliminary result of topology. Secondly non Hausdorff spaces are handled from an entirely topological point of view, which is the only rigorous and clearly motivated one, and in fact allows to get rid of all the confusion and the errors that the shaky measure theoretic “a.e.” generates. Finally the constraints on the specification of the underlying measure spaces, that our method reveals to be necessary for a correct definition of the integral, require to be met a result of measure theory, which is also established. In this way, besides the remarks on measure theory and topology, which may well be useful in many other situations, a working and precise settlement of the definition of the integral is achieved.