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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
,
Thus,
But by Corollary 1.56, item (ii), and
and a similar formula also holds for every subinterval contained in . Hence, for