ABSTRACT
But the most striking innovation in this exposition is a def inition of differential based on the integration of summants. The relation f K |5 — S'\ = 0 between summants S and S' on a cell K is an equivalence. A differential on K is just an equiv alence class of summants on K. Since the upper and lower in tegrals are invariant for equivalent summants each differential acquires an upper and lower integral. So we have the concepts of integral and integrability for differentials. Every function f on K induces an integrable differential df, the equivalence class of the summant Δ / , which satisfies (1). Moreover, ev ery integrable differential is the differential of a function. In another role functions on K act as multipliers (“differential coefficients”) of differentials. So we have the differentials fdg on K for all functions / , g on K. Since the differentials on K form a Riesz space we also have the differentials f\dg\, f(dg)+, and f (dg )~ . In addition to these we have some novel differen tials such as the unit differential ω represented by the constant summant 1.
