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Measures Defined by Gages

Published online by Cambridge University Press:  20 November 2018

Washek F. Pfeffer  and
Brian S. Thomson
Washek F. Pfeffer
Affiliation:
University of California, Davis, California 95616, U.S.A.
Brian S. Thomson
Affiliation:
Simon Fraser University, Burnaby, British Columbia, V5A 1S6
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Abstract

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Using ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].

In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

Keywords

28C1528A12
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 6 , 01 December 1992 , pp. 1303 - 1316
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Ahmed, S.I. and Pfeffer, W.F., A Riemann integral in a locally compact Hausdorff space, J. Australian Math. Soc. 41(1986),115–137.Google Scholar
2. Gardner, R.J. and F Pfeffer, W., Borel measures. In: Handbook of Set-theoretic Topology, (ed. Kunen, K. and Vaughan, J.E.), North-Holland, New York, 1984,961–1043.Google Scholar
3. Gardner, R.J. and F Pfeffer, W., Decomposability of Radon measures, Trans. American Math. Soc. 283(1984), 283–293.Google Scholar
4. McShane, E.J., A Riemann type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals, Mem. American Math. Soc. 88(1969).Google Scholar
5. Pfeffer, W.F., On the lower derivative of a set function, Canadian J. Math. 20(1968), 1489–1498.Google Scholar
6. Pfeffer, W.F., Integrals and Measures, Marcel Dekker, New York, 1977.Google Scholar
7. Roy, H.L. den, Real Analysis, Macmillan, New York, 1968.Google Scholar
8. Rudin, W., Real and Complex Analysis, McGraw-Hill, New York, 1987.Google Scholar