By hypothesis, , where denotes the Lebesgue measure. Hence, for every . In addition, . Then, for every and every , there exists a gauge on such that, for every -fine tagged partial division , with , for , we have
Consider a gauge of such that , whenever , and can assume any value in , otherwise. Then, for every -fine , we have
and we complete the proof.
Given a function , since , Theorem 1.81 holds for instead of . Then, next proposition follows easily (see, also, [70, Corollary after Theorem 5]).
Proposition 1.82:Suppose and . Assume, in addition, that is such that almost everywhere in . Then, and , for every . If, moreover, , then .
In view of Proposition 1.82, we can define equivalence classes of nonabsolute vector integrable functions.
Definition 1.83: Let us assume that . Two functions
