Let be any division of . If we take for , then the point-interval pair is a -fine tagged partial division of and, therefore, from (1.A.3) and since we can assume, without loss of generality (see comments in the paragraph before the statement), that for , we have
and the proof is complete.
Lemma 1.98:Suppose . The following properties are equivalent:
1 is absolutely integrable;
2 .
Proof. (i) (ii). Suppose is absolutely integrable. Since the variation of , , is given by
we have
(ii) (i). Suppose . We prove that the integral exists and . Given , we need to find a gauge on such that
