ABSTRACT
In the Lebesgue integration, the domain space is abstract and no topological or geometric properties, even when present, play significant roles whereas in the nonabsolute case these are utilized more fully from the start. This chapter discusses the basic properties of nonabsolute integrals, originally investigated by A. Denjoy and O. Perron in detail, from a simplified but equivalent form, due to J. Kurzweil, R. Henstock and E. J. McShane. It establishes a deep connection between the local theory of nonabsolute and the absolute (or Lebesgue type) theory via a (generalized) boundedness principle formulated by Bochner. The chapter points out how a local theory admits extensions to the global versions under natural (truncation or 'stopping time') conditions. It provides theorems and proofs for the nonabsolute integration and also includes multiple exercises that help students try themselves and perform nonabsolute integration.
