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On sequences of integrable functions

Published online by Cambridge University Press:  09 April 2009

Basil C. Rennie
Basil C. Rennie
Affiliation:
R.A.A.F. Academy, Victoria
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Let f1(x), f2(x), … be a sequence of functions belonging to the real or complex Banach space L, (see S. Banach: [1] (The results can be generalised to functions on any space that is the union of countably many sets of finite measure). We are concerned with various properties that such a sequence may have, and in particular with the more important kinds of convergence (strong, weak and pointwise). This article shows what relations connect the various properties considered; for instance that for strong convergence (i.e. ║fn — f║ → 0) it is necessary and sufficient firstly that the sequence should converge weakly (i.e. if g is bounded and measurable then f(fn(x) — f(x))g(x)dx → 0) and secondly that any sub-sequence should contain a sub-sub-sequence converging p.p. to f(x).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 3 , February 1962 , pp. 295 - 300
Copyright
Copyright © Australian Mathematical Society 1962

References

[1]Banach, S., Théorie des Opérations Linéaires Monografje Matematyczne, 1, Warsaw, (1932).Google Scholar
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[3]Rennie, B. C., On dominated convergencs, this Journal, 2, (1961) (133–136).Google Scholar
[4]du Bois Reymond, P., Untersuchungen über die Convergenx und Divergenz der Fourierschen Darstellungs formeln, Abhand. Akad. München, XII (1876) 1–103.Google Scholar