the other hand, when the idea of McShane is employed in the variational Henstock integral, one gets precisely the Bochner–Lebesgue integral. This interesting fact was proved by W. Congxin and Y. Xiabo in [47] and, independently, by C. S. Hönig in [131]. Later, L. Di Piazza and K. Musal generalized this result (see [55]). We clarify here that unlike the proof by Congxin and Xiabo, based on the Fréchet differentiability of the Bochner–Lebesgue integral, Hönig's idea to prove the equivalence between the Bochner–Lebesgue integral and the integral we refer to as Henstock–McShane integral uses the fact that the indefinite integral of a Henstock–McShane integrable function is itself a function of bounded variation and the fact that absolute Henstock integrable functions are also functions of bounded variation. In this way, the proof provided in [131] becomes simpler. We reproduce it in the next lines, since reference [131] is not easily available. We also refer to [73] for some details.
Definition 1.89: We say that a function
1 almost everywhere (i.e. for almost every ), and
2 .
With the notation of Definition 1.89, we define
Then, the space of all equivalence classes of Bochner–Lebesgue integrable functions, equipped with the norm
The next definition can be found in [239], for instance.
Definition 1.90: We say that a function
(1.A.1)
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
is a division of
