sigma-summation sigma-summation equals equals j 1 vertical-bar vertical-bar d times times of ff left-parenthesis right-parenthesis xi j left-parenthesis right-parenthesis minus minus tjt minus minus j 10 Subscript 2 Baseline equals vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals j 1 vertical-bar vertical-bar d times times e xi j left-parenthesis right-parenthesis minus minus tjt minus minus j 1 Subscript 2 Baseline 2nd Row 1st Column Blank 2nd Column equals left-bracket sigma-summation Underscript j equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts StartAbsoluteValue t Subscript j Baseline minus t Subscript j minus 1 Baseline EndAbsoluteValue squared right-bracket Superscript one half Baseline less-than delta Superscript one half Baseline left-bracket sigma-summation Underscript j equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts left-parenthesis t Subscript j Baseline minus t Subscript j minus 1 Baseline right-parenthesis right-bracket Superscript one half Baseline less-than epsilon comma EndLayout"/>
where we applied the Bessel equality. Thus,
for every
1.3.2 Basic Properties
The first result we present in this subsection is known as the Saks–Henstock lemma, and it is useful in many situations. For a proof of it, see [210, Lemma 16], for instance. Similar results hold if we replace
Lemma 1.45 (Saks–Henstock Lemma): The following assertions hold.
1 Let and . Given , let be a gauge on such that for every ‐fine ,Then, for every ‐fine , we have
2 Let and Given , let be a gauge on such that for every ‐fine ,Then, for every ‐fine , we have
Now, we define some important sets of functions.
Definition 1.46: Let
Given
By the Banach–Steinhaus theorem, the limits