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A cauchy criterion and a convergence theorem for Riemann-complete integral

Published online by Cambridge University Press:  09 April 2009

H. W. Pu
H. W. Pu
Affiliation:
Texas A&M University College Station, TexasU.S.A.
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In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 1 , February 1972 , pp. 21 - 24
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Kurzweil, J., ‘Generalized ordinary differential equations and continuous dependence on a parameter’, Czechoslovak Math. Jour. 7 (82) (1957), 418–446.CrossRefGoogle Scholar
[2]Henstock, R., Theory of Integration (Butterworths, London, 1963).Google Scholar
[3]Henstock, R., ‘A Riemann-type integral of Lebesgue power’, Canad. J. Math. 20 (1968), 79–87.CrossRefGoogle Scholar
[4]Kelley, J. L., General Topology (Van Nostrand, Princeton, 1965).Google Scholar