ABSTRACT
Let L be a nondegenerate interval in [—00, 00]. That is, (o, b) C L C [a, 6] where — 00 < a < b < 00 . So L = K = [a, 6]. In §1.9 we defined summants on I as functions on the set of all tagged cells in L. Such a summant S i has upper and lower integrals given according to (4) in §1.9 by
where S is any extension of S i to a summant on K , l i is
in L, and l i is the function on K indicating the subset L of K . So we could define a differential on L to be an equivalence class σ ι of summants S i on L with equivalence of S i and S'L defined by
for any extensions of S l , S'l to summants S , S ' on K. We could then define the upper and lower integrals of σ ι over L by
the cell summant on K indicating the set of all cells contained
(3)
under (1) with concommitant definitions of $ l <?l and integrability of σi over L, consistent with our definitions for the case L = K .
