alt="sigma-summation Underscript i equals 1 Overscript n Endscripts gamma Subscript i Baseline x Subscript i Baseline equals left-parenthesis sigma-summation Underscript i equals 1 Overscript n Endscripts gamma Subscript i Baseline right-parenthesis x 0 plus sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis sigma-summation Underscript i equals j Overscript n Endscripts gamma Subscript i Baseline right-parenthesis left-parenthesis x Subscript j Baseline minus x Subscript j minus 1 Baseline right-parenthesis comma n element-of double-struck upper N period"/>
Hence, taking
where we applied the Saks–Henstock lemma (Lemma 1.45) to obtain
for every
A proof of the next proposition follows similarly as the proof of Theorem 1.66.
Proposition 1.67: Let be any interval of the real line and , with . Consider functions and of locally bounded variation. Assume that is locally Perron–Stieltjes integrable with respect to , that is, the Perron–Stieltjes integral exists, for every compact interval . Assume, further, that , defined by
is also of locally bounded variation. Then, the Perron–Stieltjes integrals and exist and
(1.11)
Yet another substitution formula for Perron–Stieltjes integrals, borrowed from [72, Theorem 11], is brought up here and, again, another interesting trick provided by Professor Hönig is used in its proof. Such substitution formula will be used in Chapter 3 in order to guarantee the existence of some Perron–Stieltjes integrals. As a matter of fact, the corollary following Theorem 1.68 will do the job.
Theorem 1.68: Consider functions , , , that is,
and assume that . Thus, if and only if , in which case, we have
(1.12)
Proof. By hypothesis,