An Alternative Proof of a Theorem on the Lebesgue Integral

Abstract

By (2), F(x)= \f'\ is a.c. Hence 3<5,>O such that if {[ar, br)} is a Ja finite set of non-overlapping intervals and T.(b, — ar)O such that if | br — ar | <<52, then \Abr)-Kar) | <\E (5) Now take 8 = min^ , S2, I, e/SK), and choose intervals [ar, br] to satisfy (3). The sum Z \f(b^-f{ar) | may be divided into three parts, by putting \ fibr)-f(ar)\ into S, if/isa.c. in [ar, br], Z2 if/is not a.c. in [ar, b,] and \f(br)-f(ar) \^K(br-ar), S3 if/is not a.c. in [ar, br] and \f(br)-f(ar) \>K(br-ar). Now if/is a.c. in [ar, br], then by (1), ,)-/(«,) | = I f"f ^ T I / ' I = I F{br)F(a,) |. I Jar Jar E.M.S.—H J