Upper and Lower Lebesgue Integrals

On functions of bounded ω-variation, II

“…A function F(a;) defined on [a, J] and belonging to the class % is said to be absolutely continuous above relative to co, AC-co above, if for s > 0 there exists d > 0 such that for any elementary system / in [a, b] with I a < d the relation al < s holds. It is said to be absolutely continuous below relative to co, AC-co below, if the relation al > -e holds whenever I u < d. [2] On functions of bounded cu-variation, II This definition is analogous to the definition in [3] Throughout the paper we shall consider only those functions F(x) of the class % for which F(x+) and F(x-), x e E-S, are finite.…”

“…This definition is analogous to the definition in [3] Throughout the paper we shall consider only those functions F(x) of the class % for which F(x+) and F(x-), x e E-S, are finite. li a = x lt x' n = box a -x lt x' n <b or a <x x ,x' n = b then it can be similarly shown that al 2g G, a fixed constant independent of/.…”