minus t Subscript i minus 1 Baseline right-parenthesis minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column less-than-or-slanted-equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts StartAbsoluteValue vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis xi i left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis minus vertical-bar vertical-bar vertical-bar vertical-bar integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t EndAbsoluteValue plus StartAbsoluteValue sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue 3rd Row 1st Column Blank 2nd Column less-than-or-slanted-equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus times times of ff left-parenthesis right-parenthesis xi i left-parenthesis right-parenthesis minus minus tit minus minus i 1 integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t plus StartAbsoluteValue sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis ti of ff tilde left-parenthesis right-parenthesis t minus minus i 1 minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue period EndLayout"/>
By the definition of , we may take such that the last summand in (1.A.4) is smaller than . Because , we may take a gauge such that for every -fine , the first summand in (1.A.4) is also smaller than (and we may suppose that the points chosen for the second summand are the points of the -fine tagged division ).
The next result is a consequence of the fact that and Lemmas 1.97 and 1.98. A proof of it can be found in [132, Theorem 10.3].
Corollary 1.99: All functions of are absolutely.integrable
The reader can find a proof of the next lemma in [35, Theorem 9].
Lemma 1.100: All functions of are.measurable
Finally, we can prove the following inclusion.
Theorem 1.101: .
Proof. The result follows from the facts that all functions of and, hence, of are measurable (Lemma 1.100), and all functions of are absolutely integrable by Corollary 1.99.
As we mentioned before, the inclusion always holds. However, when is an infinite dimensional Banach space, then for sure , as shown by the next result due to C. S. Hönig (personal communication by him to his students in 1990 at the University of São Paulo) and presented in [73].
Proposition 1.102 (Hönig): If is an infinite dimensional Banach space, then there exists .
Proof. Let denote the dimension of . If , then the Theorem of Dvoretsky–Rogers (see [60] and also [57]) implies there exists a sequence in which is summable but not absolutely summable. Thus, if we define a function
