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A converse to Lebesgue's dominated convergence theorem

Published online by Cambridge University Press:  09 April 2009

Dwight B. Goodner
Dwight B. Goodner
Affiliation:
Florida State University
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Let (X, B, m) be a measure space and let f(x) be a real-valued or complex-valued measurable function on X. A non-negative measurable function s(x) will be said to dominate f(x) provided |f(x)| ≦ s(x) for almost all x in X. The function s(x) will be said to dominate the sequence {f(x)}n∈N, N = {1, 2,…}, provided it dominates each fn(x) in the sequence. Unless otherwise specified, each integral will be over X with respect to m.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 4 , November 1966 , pp. 477 - 479
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Lebesgue, H., ‘Sur l'intégration des fonctions discontinues’, Ann. Ecole Norm. 27 (1910), 361–450.Google Scholar
[2]Rennie, B. C., ‘On dominated convergence’, Jour. Australian Math. Soc. 2 (1961–1962), 133–136.CrossRefGoogle Scholar
[3]Saks, S., Theory of the integral, 2nd ed., Monografje Matematyczne 7, Warsaw, 1937.Google Scholar