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and the following integration by parts formula holds
Corollary 1.62: Consider functions and . Then, and equalities (1.6) and (1.7) hold.
The next two theorems generalize Corollary 1.62. For their proofs, the reader may want to consult [72].
Theorem 1.63: Consider . If respectively, , then respectively, and both (1.6) and (1.7) hold.
The next result is a Substitution Formula for Perron–Stieltjes integrals. A similar result holds for Riemann–Stieltjes integrals. For a proof of it, see [72, Theorem 11].
Theorem 1.64: Consider functions , , and . Let
Then, if and only if , in which case, we have
(1.8)
(1.9)
Using Theorem 1.53, one can prove the next corollary. See [72, Corollary 8]. From now on,
Corollary 1.65: Consider functions , , and , and define
Then, and equality (1.8) and inequality (1.9) hold.
Another substitution formula for Perron–Stieltjes integrals is presented next. Its proof uses a very nice trick provided by Professor C. S. Hönig while advising M. Federson's Master Thesis. Such result is borrowed from [72, Theorem 12].
Theorem 1.66: Consider functions , and , that is,
Then, , if and only if and
(1.10)
Proof. Since
Taking approximated sums for
