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But by Corollary 1.56, item (ii),
and a similar formula also holds for every subinterval contained in
since for every
The proof is then complete.
A proof of the next result, borrowed from [72, Theorem 8], follows from the definitions of the integrals.
Theorem 1.59: Let and . If is bounded, then and
(1.5)
If, in addition, , then .
Corollary 1.60: Suppose with and . Then, and (1.5) holds.
Proof. By Theorem 1.49,
The next corollaries follow from Theorems 1.49 and 1.53.
Corollary 1.61: Suppose with and . Then, , and we have
