UPPER AND LOWER HENSTOCK INTEGRALS

Abstract

The Riemann integral can be defined in terms of upper and lower Riemann sums. In this paper, we show how the Henstock integral can also be defined in a similar manner. A natural generalization of the Riemann integral is the Henstock integral (see, for example, [2]), which also includes the Lebesgue integral and the Newton integral. The Henstock integral is also known as the Kurzweil or Kurzweil-Henstock integral. Since the Riemann integral is defined for bounded functions only, upper and lower Riemann sums always exist. However this is no longer the case for the Henstock integral, since Henstock integrable functions are not necessarily bounded. The essential idea in the definition of the Henstock integral is that when forming Riemann sums we do not use all the points in each interval of a partition as it is done in the Riemann integral, rather we use only certain designated points in the interval. This is done by introducing a positive function δ(x), and using only those points ξ in [u, v] for which [u, v] ⊂ (ξ−δ(ξ), ξ+δ(ξ)). Keeping to such ξ, a kind of upper and lower sums can now be define. In this paper, we identify the designated points directly by introducing a contraction of intervals into their subsets, which serves the same purpose as the positive function δ(ξ) in the Henstock integral, and thus define corresponding upper and lower sums and in turn the upper and lower Henstock integrals. When the two integrals are equal, we obtain the Henstock integral. We recall that given δ(ξ) > 0, a partition D of [0, 1] given by x0 = 0 < x1 < · · · < xn = 1, ξ1, ξ2, · · · , ξn,