The Henstock-Kurzweil integral with integrators of unbounded variation

Abstract

In 1936, L.C. Young proved that f is Riemann integrable with respect to g on [a,b], if f and g are of bounded p and q-variations, respectively, where 1/p+1/q>1 and f and g do not have common discontinuity points. Two years later, he was able to drop the condition on common discontinuity for his new integral (called Young integral). The Young integral is defined by the Moore-Pollard approach. In other words, the integral is defined by way of refinements of partitions and the integral is the Moore-Smith limit of the Riemann-Stieltjes sums using the directed set of partitions. However, modified Riemann-Stieltjes sums involving g(x+) and g(x-) are used in Young integrals. Furthermore, he generalized his result to functions of bounded phi-variation. The Young approach is rather involved. In this thesis, we shall use the Henstock-Kurzweil approach to handle the Young integral. In this approach, non-uniform meshes are used in Riemann sums. Non-uniform mesh is able to handle highly oscillatory integrands and integrators. This approach is more intuitive. We shall also generalize the results in one-dimesional case to the two-dimensional case.