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

Appendix 1.A: The McShane Integral

The integrals introduced by J. Kurzweil [152] and independently by R. Henstock [118] in the late 1950s are equivalent to the restricted Denjoy integral and the Perron integral for integrands taking values in (see [108], for instance). In particular, the definitions of the so-called ”Kurzweil–Henstock” integrals are based on Riemannian sums, and are therefore easy to deal with even by undergraduate students. Not only that, but the Kurzweil–Henstock–Denjoy–Perron integral encompasses the integrals of Newton, Riemann, and Lebesgue.

      In 1969, E. J. McShane (see [173, 174]) showed that a small change in the subdivision process of the domain of integration within the Kurzweil–Henstock (or Perron) integral leads to the Lebesgue integral. This is a very nice finding, since now the Lebesgue integral can be taught by presenting its Riemannian definition straightforwardly and, then, obtaining immediately some very interesting properties such as the linearity of the Lebesgue integral which comes directly from the fact that the Riemann sum can be split into two sums. The monotone convergence theorem for the Lebesgue integral is another example of a result which is naturally obtained from its equivalent definition due to McShane.

      The Kurzweil integral and the variational Henstock integral can be extended to Banach space-valued functions as well as to the evaluation of integrands over unbounded intervals. The extension of the McShane integral, proposed by R. A. Gordon (see [107]) to Banach space-valued functions, gives a more general integral than that of Bochner–Lebesgue. As a matter of fact, the idea of McShane into the definition due to Kurzweil enlarges the class of Bochner–Lebesgue integrals.

      On

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