Article 16

Integration by parts for the Foran integral

Real Analysis Exchange 17 (1992), 402-404.

Mathematical reviews subject classification: 26A39.

Abstract

We give the following formula for integration by parts for the Foran integral:

Let F, g:[a,b] → R be such that F ∈ VB and g is a Foran integrable (short (F)-integrable) function on [a,b].

Then F(x)g(x) is Foran integrable, and denoting the Foran indefinite integral of g by G, we have

$(F) \int_a^b F(x)g(x)\,dx = g(b)F(b) - G(a)F(a) - (S) \int_a^b G(x)\,dF(x).$

(Here (S) denotes the Riemann Stieltjes integral.)

This result is an extension of Theorem 2.5 of S. Saks (Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937).