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Completely additive measure and integration

Author: Alan McK. Shorb
Journal: Proc. Amer. Math. Soc. 53 (1975), 453-459
MSC: Primary 28A10; Secondary 02H25
MathSciNet review: 0382578
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Abstract: This paper is an extension of the efforts to cast the theory of measure and integration into the framework of nonstandard analysis, begun by Robinson [7, particularly Theorem 3.5.2], and continued by Bernstein and Wattenberg, Loeb and Henson. The principal result, Theorem 3, is: There exists a completely additive measure function defined on all subsets of $ R$ which nearly agrees with Lebesgue measure and is nearly translation invariant on bounded sets. Its integral is defined for all sets and functions, and nearly agrees with the Lebesgue integral.

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  • [1] Allen R. Bernstein, A nonstandard integration theory for unbounded functions, Z. Math. Logik Grundlagen Math. 20 (1974), 97-108. MR 0360972 (50:13419)
  • [2] Allen R. Bernstein and Peter A. Loeb, A nonstandard integration theory for unbounded functions, Victoria Symposium on Nonstandard Analysis, Lecture Notes in Math., vol. 369, Springer-Verlag, New York, 1974. MR 0492167 (58:11313)
  • [3] A. R. Bernstein and F. Wattenberg, Nonstandard measure theory, Applications of Model Theory to Algebra, Analysis and Probability (Internat. Sympos., Pasadena, Calif., 1967), Holt, Rinehart and Winston, New York, 1969, pp. 171-185. MR 40 #287. MR 0247018 (40:287)
  • [4] F. Hausdorff, Grundzüge der Mengenlehre, Veit, Leipzig, 1914; photographic reproduction, Chelsea, New York, 1949. MR 11, 88. MR 0031025 (11:88d)
  • [5] C. Ward Henson, On the representation of measures, Trans. Amer. Math. Soc. 172 (1972), 437-446. MR 47 #3631. MR 0315082 (47:3631)
  • [6] Peter A. Loeb, A nonstandard representation of measurable spaces, $ {L^\infty }$ and $ L_\infty ^{\ast}$, Contributions to Nonstandard Analysis, W. A. J. Luxemburg and A. Robinson (Editors), North-Holland, Amsterdam, 1972, pp. 65-80. MR 0482128 (58:2215)
  • [7] Abraham Robinson, Non-standard analysis, North-Holland, Amsterdam, 1966. MR 34 #5680. MR 0205854 (34:5680)
  • [8] H. L. Royden, Real analysis, Macmillan, New York, 1963. MR 27 #1540. MR 0151555 (27:1540)
  • [9] A. M. Shorb, A completely additive nonstandard measure function on $ R$, Notices Amer. Math. Soc. 20 (1973), A-32. Abstract #701-02-9.

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Article copyright: © Copyright 1975 American Mathematical Society

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