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* *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*.

**[1]**Allen R. Bernstein,*A non-standard integration theory for unbounded functions*, Z. Math. Logik Grundlagen Math.**20**(1974), 97–108. MR**0360972****[2]**Allen R. Bernstein and Peter A. Loeb,*A non-standard integration theory for unbounded functions*, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972) Springer, Berlin, 1974, pp. 40–49. Lecture Notes in Math., Vol. 369. MR**0492167****[3]**Allen R. Bernstein and Frank 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**0247018****[4]**Felix Hausdorff,*Grundzüge der Mengenlehre*, Chelsea Publishing Company, New York, N. Y., 1949 (German). MR**0031025****[5]**C. Ward Henson,*On the nonstandard representation of measures*, Trans. Amer. Math. Soc.**172**(1972), 437–446. MR**0315082**, 10.1090/S0002-9947-1972-0315082-2**[6]**Peter A. Loeb,*A non-standard representation of measurable spaces, 𝐿_{∞}, and 𝐿*_{∞}*, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), North-Holland, Amsterdam, 1972, pp. 65–80. Studies in Logic and Found. Math., Vol. 69. MR**0482128****[7]**Abraham Robinson,*Non-standard analysis*, North-Holland Publishing Co., Amsterdam, 1966. MR**0205854****[8]**H. L. Royden,*Real analysis*, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR**0151555****[9]**A. M. Shorb,*A completely additive nonstandard measure function on*, Notices Amer. Math. Soc.**20**(1973), A-32. Abstract #701-02-9.

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0382578-5

Article copyright:
© Copyright 1975
American Mathematical Society