alt="upper L left-parenthesis upper X comma upper Y right-parenthesis"/>. Then, by the Uniform Boundedness Principle, The next result concerns the existence of Perron–Stieltjes integrals. A proof of its item (i) can be found in [210, Theorem 15]. A proof of item (ii) follows similarly as the proof of item (i). See also [212, Proposition 7].
Theorem 1.47: The following assertions hold.
1 If and , then .
2 If and , then we have .
The following consequence of Theorem 1.47 will be used later in many chapters. The inequalities follow after some calculations. See, for instance, [210, Proposition 10].
Corollary 1.48: The following assertions hold.
1 If and , then the Perron–Stieltjes integral exists, and we haveSimilarly, if and is nondecreasing, then
2 If and , then the Perron–Stieltjes integral exists, and we have
The next result, borrowed from [74, Theorem 1.2], gives us conditions for indefinite Perron–Stieltjes integrals to be regulated functions.
Theorem 1.49: The following assertions hold:
1 if and , then ;
2 if and , then .
Proof. We prove (i). Item (ii) follows similarly. For item (i), it is enough to show that
because, in this case, the equality
follows in an analogous way. By hypothesis,
Now, let
If
Thus,
