A Remark on the Change of Variable Theorem for the Riemann Integral

Alexander Kuleshov

doi:10.3390/math9161899

Abstract

In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted while the change of variable formula still holds.

Highlights

Throughout this paper, we denote [ a, b] as the closed interval connecting the points a, b ∈ R, and denote R[ a, b] as the class of all Riemann-integrable real functions on [ a, b]
The main goal of this note is to abandon the requirement of boundedness of the function f on [c, d] := G ([α, β]) in Theorem 2
The condition for the boundedness of the function f on [ G (α), G ( β)] is essential for the existence of the integral on the left-hand side of (2) and does not follow from other conditions of the theorem, which are shown by the example at the end of [3]

Summary

Introduction

Throughout this paper, we denote [ a, b] as the closed interval connecting the points a, b ∈ R, and denote R[ a, b] as the class of all Riemann-integrable real functions on [ a, b]. In 1961, Kestelman (see [1]) first proved the following fundamental theorem for the Riemann integral. ( f ◦ G ) g ∈ R[α, β] and the following change of variable formula holds: GZ( β). The main goal of this note is to abandon the requirement of boundedness of the function f on [c, d] := G ([α, β]) in Theorem 2. The condition for the boundedness of the function f on [ G (α), G ( β)] is essential for the existence of the integral on the left-hand side of (2) and does not follow from other conditions of the theorem, which are shown by the example at the end of [3].

The Main Result

Some applications