Algebras of Riemann Integrable Functions

In the present paper we shall consider from two different standpoints the real Riemann integrable functions on a bounded closed interval / contained in En. Our results will be extendable to the complex case by standard arguments. (1) In §1 we designate the real valued Riemann integrable functions by R(I). If R(I) is normed by ||/|| =supx6j |/(x)| then it is known that R(I) is a Banach algebra. We show that R(I) can be characterized as a B(S, 2) space [l] which consists of all uniform limits of real finite linear combinations of characteristic functions of sets in I having Jordan content. This characterization immediately yields a representation of the dual space of R(I). (2) We consider in §2 a collection of equivalence classes of functions in R(I) : we define RX(I) to be the collection of classes such that /i and/2 are in the same class if and only iifi—fi vanishes everywhere except on a set of Lebesgue measure zero. If [f] is an element of RX(I) we define the norm || [f]|| as follows. Let N designate the subsets of I of Lebesgue measure zero. Then if / E \f] let ||[/]|| = infse¡v supxîs |/(x)| =||/||o». It is obvious that this norm is induced by the norm of LX(I). It is also clear that Rm(I) is isometrically isomorphic to a normed subalgebra of LX(I). Designating the Boolean ring of null Jordan sets by $ C\ N, we show that i?M(/) = R(I)/B(I, ¿JfW) after first establishing that B(l, ¿JHA/) (interpreted in the B(S, 2) sense) is a closed ideal in R(I). It is a pleasure to acknowledge the advice of Professor Fred B. Wright and Professor F. Quigley.