Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

      Again, we explicit the “name” of the integral we are dealing with, whenever we believe there is room for ambiguity.

      As we mentioned earlier, when only real-valued functions are considered, the Lebesgue integral is equivalent to a modified version of the Kurzweil–Henstock (or Perron) integral called McShane integral. The idea of slightly modifying the definition of the Kurzweil–Henstock integral is due to E. J. McShane [173, 174]. Instead of taking tagged divisions of an interval , McShane considered what we call semitagged divisions, that is,

      is a division of and, to each subinterval , with , we associate a point called “tag” of the subinterval . We denote such semitagged division by and, by , we mean the set of all semitagged divisions of the interval . But what is the difference between a semitagged division and a tagged division? Well, in a semitagged division , it is not required that a tag belongs to its associated subinterval . In fact, neither the subintervals need to contain their corresponding tags. Nevertheless, likewise for tagged divisions, given a gauge of , in order for a semitagged division to be -fine, we need to require that

      This simple modification provides an elegant characterization of the Lebesgue integral through Riemann sums (see [174]).

Let us denote by the space of all real-valued Kurzweil–McShane integrable functions , that is, is integrable in the sense of Kurzweil with the modification of McShane. Formally, we have the next definition which can be extended straightforwardly to Banach space-valued functions.

      Definition 1.91: We say that is Kurzweil–McShane integrable, and we write if and only if there exists such that for every , there is a gauge on such that

      whenever is -fine. We denote the Kurzweil–McShane integral of a function by .

      The following inclusions hold

      Moreover,