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The next theorem is due to C. S. Hönig (see [129]), and it concerns multipliers for Perron–Stieltjes integrals.
Theorem 1.57:Suppose and . Then, and Eqs. (1.3) and (1.4) hold.
Since and , it is immediate that if and , then . As a matter of fact, the next result gives us information about the multipliers for the Henstock vector integral. See [72, Theorem 7].
Theorem 1.58:Assume that and . Then, and equalities (1.3) and (1.4) hold.
Proof. Since , is continuous by Theorem 1.49. Thus, given , there exists such that
whenever , where . Moreover, there is a gauge on , with for , such that for every ‐fine ,
