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In this chapter we investigate the space of the Henstock-Dunford integral function. For each function which is Henstock-Dunford integral can be constructed by an operator. For any Henstock-Dunford integrable function f, an operator can be constructed such that it is both a continuous linear operator and a weakly compact operator. Furthermore, the adjoint operator is continuous and weakly compact linear operator for the case of Henstock-Dunford integral. In the end, we introduce Pettis integrability-type property for HKP-integrals. We discuss several necessary conditions that \(\mathcal {X}\) has HKP-integrability property for weak Baire measure. The necessary and sufficient conditions is that the indefinite integral of any Henstock-Kurzweil-Pettis (respectively, Denjoy-Pettis) integrable function with values in a fixed Banach space have separable ranges are discussed.

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Henstock-Dunford and Henstock-Pettis Integrable Functions

  • 章节

  • 发表日期：

    2025年

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  • Bipan Hazarika,
  • Hemanta Kalita

摘要

In this chapter we investigate the space of the Henstock-Dunford integral function. For each function which is Henstock-Dunford integral can be constructed by an operator. For any Henstock-Dunford integrable function f, an operator can be constructed such that it is both a continuous linear operator and a weakly compact operator. Furthermore, the adjoint operator is continuous and weakly compact linear operator for the case of Henstock-Dunford integral. In the end, we introduce Pettis integrability-type property for HKP-integrals. We discuss several necessary conditions that \(\mathcal {X}\) has HKP-integrability property for weak Baire measure. The necessary and sufficient conditions is that the indefinite integral of any Henstock-Kurzweil-Pettis (respectively, Denjoy-Pettis) integrable function with values in a fixed Banach space have separable ranges are discussed.