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On the integrals of Perron type Lee, Cheng-Ming

Abstract

Perron's method of defining a process of integration is through the use of major and minor functions. Many authors have adopted this method to define various integrals. In Chapter I, we give a very general abstract theory by first defining an abstract "derivate system" and then the corresponding Perron integral. We show that this unifies all the integral theories of Perron type (of first order) known to us, in addition the abstract theories of Pfeffer [26] and of Romanovski [29] are contained in our theory as particular cases. Chapter II is devoted mainly to the study of Burkill's C[sub n]P - integral. We know that the C[sub n]P - integral is based on the theorem that if M is C[sub n] - continuous in [a,b] , C[sub n] DM(x) ≥ 0 almost everywhere and C[sub n] DM(x) > - ∞ nearly everywhere in [a,b] , then M is monotone increasing in [a,b] . Burkill's original proof of this, [6] , contains an error and we give it a new and correct proof. We also give a correct proof of Sargent's theorem that if a function is C[sub n]P - integrable, then it is C[sub n]D - integrable, [32] ; the original proof contains a gap. A scale of symmetric CP - integrals and a scale of approximately mean-continuous integrals are obtained in Chapter III and in Chapter IV, respectively. The first one is more general than Burkill's CP - scale, while the second one is more general than the GM - scale defined by Ellis. Some other comparisons of various integrals are also given.

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