A Note on Henstock-Itô's Non-Stochastic Integral

Abstract

It is well-known that the generalized Riemann approach using non-uniform mesh has given rise to integrals which are more general than the Riemann integral. In this note, motivated by earlier studies on stochastic integrals, we consider special interval-point pair in defining the Riemann sum, where the point (or the tag) is the left-hand point of the interval. We show that this approach in fact is equivalent to the Lebesgue integral. Hence this simplifies McShane's construction of interval-point pair. Moreover, by restricting to the tag as the left hand point of the interval, the integration-by-substitution and by-parts formulae become easy consequences of the definition.