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Singular Integrals are Perron Integrals of a Certain Type

  • W. F. Pfeffer (a1)

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In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.

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References

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1. Alexandroff, P. S. and Urysohn, P. S., Mémoire sur les espaces topologique compacts, Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. I 14 (1929), No. 1, 196.
2. Bogdanowicz, W. M., A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 492498.
3. Hayes, C. A. and Pauc, C. Y., Full individual and class differentiation theorems in their relations to halo and Vitali properties, Can. J. Math. 7 (1955), 221274.
4. Kelley, J. L., General topology (Van Nostrand, New York, 1955).
5. Mikhlin, S. G., Multidimensional singular integrals and integral equations (Pergamon Press, New York, 1965).
6. Pfeffer, W. F., On the lower derivative of a set function, Can. J. Math. 20 (1968), 14891498.
7. Pfeffer, W. F., An integral in topological spaces. I, J. Math. Mech. 18 (1969), 953972.
8. Pfeffer, W. F., An integral in topological spaces, Research announcement, Bull. Amer. Math. Soc. 75 (1969), 433439.
9. Zemanian, A. H., Distribution theory and transform analysis (McGraw-Hill, New York, 1965).
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Singular Integrals are Perron Integrals of a Certain Type

  • W. F. Pfeffer (a1)

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