sums concerning ‐fine tagged divisions of an interval , we bring up an important result which guarantees the existence of a ‐fine tagged division for a given gauge . This result is known as Cousin Lemma, and a proof of it can be found in [120, Theorem 4.1].
Lemma 1.36 (Cousin Lemma):Given a gauge of , there exists a ‐fine tagged division of .
Definition 1.37: We say that is Kurzweil‐integrable (or Kurzweil integrable with respect to), if there exists such that for every , there is a gauge on such that for every ‐fine ,
In this case, we write and .
Analogously, we define the Kurzweil integral of with respect to a function .
Definition 1.38: We say that is Kurzweilintegrable (or Kurzweil integrable with respect to), if there exists such that given , there is a gauge on such that
whenever is ‐fine. In this case, we write and .
Suppose the Kurzweil vector integral exists. Then, we define
An analogous consideration holds for the Kurzweil vector integral .
If the gauge in the definition of is a constant function, then we obtain the Riemann–Stieltjes integral , and we write . Similarly, when we consider only constant gauges in the definition of , we obtain the Riemann–Stieltjes integral
