2. Preliminary definitions and theorems. Throughout this paper all integrals considered will be Hellinger [1i type limits of the appropriate sums (the definitions, theorems and proofs of this paper can be extended to "many valued" functions). Thus, if K is a real valued function of subintervals of the number interval [a, b], the existence of f[a,bjK(I) is necessary and sufficient for the existence of f1K(J) for each subinterval I of [a, b]. Furthermore, the function F of subintervals of [a, b], defined by F(I) =fK(J), is additive. If K is a real valued function of subintervals of [a, b], then the statement that K is 1-bounded on [a, b] means that there is a subdivision D of [a, b] such that the set of sums EEK(I), where E is a refinement of D and the sum is taken over all intervals I of E, is bounded. This implies that if I is an interval in a refinement of D, then the set of sums EQ K(J), where Q is a subdivision of I and the sum is taken over all intervals J of Q, has a least upper bound L(I) and a greatest lower bound G(I). We now see that if each of R and T is a refinement of D, and S is a refinement of each of R and T, then ER G(I)-< Es G(I) -< s K(I) ? Es L(I) ? E2T L(I). From this it follows that each of f(a,b]G(I) and f[a,b]L(I) exists, that f[a,b]G(I) < f[a,b]L(I), and that f(a,b]K(I) exists if and only if f[a,b]G(J) f[a,b]L(I). We state without proof a theorem of Kolmogoroff [2].