right-parenthesis t d t less-than infinity period"/>
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,