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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 are equivalent if and only if their indefinite integrals coincide, that is, . By and we mean, respectively, the spaces of equivalence classes of functions of and of , and we endow these spaces with an Alexiewicz-type norm
From Example 1.74, we know that although may belong to the same equivalence class, that is, for all , one cannot conclude that almost everywhere in . It is known, however, that the space of all equivalence classes of real-valued Perron integrable functions , equipped with the usual
