The approximation of Banach space valued nonabsolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximate in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact. It is also proved that if the target Banach space X does not contain any isomorphic copy of c_0, then the range of the integral of each X valued Denjoy-Khintchine-Pettis integrable function is norm relatively compact.

BONGIORNO, B., DI PIAZZA, L., MUSIAL, K. (2008). Approximation by step functions of Banach space valued nonabsolute integrals. GLASGOW MATHEMATICAL JOURNAL, 50(3), 583-593 [10.1017/S0017089508004448].

Approximation by step functions of Banach space valued nonabsolute integrals.

BONGIORNO, Benedetto;DI PIAZZA, Luisa;
2008

Abstract

The approximation of Banach space valued nonabsolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximate in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact. It is also proved that if the target Banach space X does not contain any isomorphic copy of c_0, then the range of the integral of each X valued Denjoy-Khintchine-Pettis integrable function is norm relatively compact.
Settore MAT/05 - Analisi Matematica
BONGIORNO, B., DI PIAZZA, L., MUSIAL, K. (2008). Approximation by step functions of Banach space valued nonabsolute integrals. GLASGOW MATHEMATICAL JOURNAL, 50(3), 583-593 [10.1017/S0017089508004448].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/40046
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 7
social impact