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and the proof is complete.
With Theorem 1.49 at hand, the next corollary follows immediately.
Corollary 1.50: The following statements hold:
1 If and , then .
2 If and , then .
Next, we state a uniform convergence theorem for Perron–Stieltjes vector integrals. A proof of such result can be found in [210, Theorem 11].
Theorem 1.51: Let and , , be such that the Perron–Stieltjes integral exists for every and uniformly in . Then, exists and
We finish this subsection by presenting a Grönwall‐type inequality for Perron–Stieltjes integrals. For a proof of it, we refer to [209, Corollary 1.43].
Theorem 1.52 (Grönwall Inequality): Let be a nondecreasing left‐continuous function, and . Assume that is bounded and satisfies
Then,
Other properties of Perron–Stieltjes integrals can be found in Chapter 2, where they appear within the consequences of the main results presented there.
1.3.3 Integration by Parts and Substitution Formulas
The first result of this section is an Integration by Parts Formula for Riemann–Stieltjes integrals. It is a particular consequence of Proposition 1.70 presented in the end of this section. A proof of it can be found in [126, Theorem II.1.1].
Theorem 1.53 (Integration by Parts): Let be a BT. Suppose
1 either and ;
2 or and .
Then, and , that is, the Riemann–Stieltjes integrals and exist, and moreover,
Next, we state a result which is not difficult to prove using the definitions involved in the statement. See [72, Theorem 5]. Recall that the indefinite integral of a function
Theorem 1.54: Suppose and is bounded. Then and
(1.3)
If, in addition, , then .
By Theorem 1.47, the Perron–Stieltjes integral
Corollary 1.55: Let and be such that . Then, and (1.3) holds.
A second corollary of Theorem 1.54 follows by the fact that Riemann–Stieltjes integrals are special cases of Perron–Stieltjes integrals. Then, it suffices to apply Theorems 1.49 and 1.53.
Corollary 1.56: Suppose the following conditions hold:
1 either and , with ;
2 or and .
Then, , equality (1.3) holds, and we have
(1.4)