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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,
