Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

any nondegenerate subinterval of . Hence, . Then, the Riemann integral exists and

(1.15)

where we applied the Fundamental Theorem of Calculus for the Riemann integral in order to obtain the last equality. Thus, replacing (1.15) in (1.14), we obtain

(1.16)

Now, in view of (1.13), it remains to prove that the Perron–Stieltjes integral exists and

(1.17)

      Let be the gauge on from the definition of . Take , and for every , let be such that if , then, by the Straddle Lemma (Lemma 1.86), we have

(1.18)

Fix . We now define a gauge on by , for every . Hence, for every -fine , we have