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On dominated convergence

Published online by Cambridge University Press:  09 April 2009

Basil C. Rennie
Basil C. Rennie
Affiliation:
University of Adelaide.
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The great theorem on convergence of integrals is due in its usual form to Lebesgue [2] though its origins go back to Arzela [1]. It says that the integral of the limit of a sequence of functions is the limit of the integrals if the sequence is dominated by an integrable function. This paper investigates the converse problem — if we know that we may take limits under the integral sign, then what can we say about the convergence? The answer is found for functions of a real variable, but it is easily extended to any space with a countably additive measure. Finally the result is illustrated by an application to Fourier series.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 2 , October 1961 , pp. 133 - 136
Copyright
Copyright © Australian Mathematical Society 1961

References

[1]Arzela, C., Sulla integrazione per serie, Rom. Acc. L. Rend., Vol 1, (1885) 532–537, 566–569.Google Scholar
[2]Lebesgue, H., Sur l'intégration des fonctions discontinues, Ann. Ecole Norm. (3) 27, (1910) 361–450.Google Scholar
[3]Titchmarsh, E. C., The theory of functions, Oxford University Press, (1932)Google Scholar