Star-finite representations of measure spaces

Author:
Robert M. Anderson

Journal:
Trans. Amer. Math. Soc. **271** (1982), 667-687

MSC:
Primary 03H05; Secondary 28D05, 60A10

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654856-1

MathSciNet review:
654856

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Abstract | References | Similar Articles | Additional Information

Abstract: In nonstandard analysis, -finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. -finite models are particularly useful in building new models of economic or probabilistic processes.

It is natural to ask what standard models can be obtained from these -finite models. In this paper, we show that a rich class of measure spaces, including the Radon spaces, are measure-preserving images of -finite measure spaces, using a construction introduced by Peter A. Loeb [**15**]. Moreover, we show that a number of measure-theoretic constructs, including integrals and conditional expectations, are naturally expressed in these models. It follows that standard models which can be expressed in terms of these measure spaces and constructs can be obtained from -finite models.

**[1]**Robert M. Anderson,*A non-standard representation for Brownian motion and Itô integration*, Israel J. Math.**25**(1976), no. 1-2, 15–46. MR**0464380**, https://doi.org/10.1007/BF02756559**[2]**-,*Star-finite probability theory*, Ph.D. Dissertation, Yale Univ., New Haven, Conn., 1977.**[3]**-,*Strong core theorems with nonconvex preferences*, Cowles Foundation Discussion Paper No. 590, Yale Univ., New Haven, Conn., 1981.**[4]**Robert M. Anderson and Salim Rashid,*A nonstandard characterization of weak convergence*, Proc. Amer. Math. Soc.**69**(1978), no. 2, 327–332. MR**0480925**, https://doi.org/10.1090/S0002-9939-1978-0480925-X**[5]**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****[6]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[7]**Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR**1009162**

Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part II*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR**1009163**

Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part III*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. MR**1009164****[8]**Lester L. Helms,*Mean convergence of martingales*, Trans. Amer. Math. Soc.**87**(1958), 439–446. MR**0094841**, https://doi.org/10.1090/S0002-9947-1958-0094841-0**[9]**C. Ward Henson,*On the nonstandard representation of measures*, Trans. Amer. Math. Soc.**172**(1972), 437–446. MR**0315082**, https://doi.org/10.1090/S0002-9947-1972-0315082-2**[10]**Douglas N. Hoover,*Probability logic*, Ann. Math. Logic**14**(1978), no. 3, 287–313. MR**510234**, https://doi.org/10.1016/0003-4843(78)90022-0**[11]**H. Jerome Keisler,*Hyperfinite model theory*, Logic Colloquium 76 (Oxford, 1976) North-Holland, Amsterdam, 1977, pp. 5–110. Studies in Logic and Found. Math., Vol. 87. MR**0491155****[12]**Peter A. Loeb,*A nonstandard representation of measurable spaces and 𝐿_{∞}*, Bull. Amer. Math. Soc.**77**(1971), 540–544. MR**0276748**, https://doi.org/10.1090/S0002-9904-1971-12745-3**[13]**-,*A nonstandard representation of measurable spaces*, ,*and*, Contributions to Non-standard Analysis (W. A. J. Luxemburg and A. Robinson, editors), North-Holland, Amsterdam, 1972, pp. 65-80.**[14]**Albert Hurd and Peter Loeb (eds.),*Victoria Symposium on Nonstandard Analysis*, Lecture Notes in Mathematics, Vol. 369, Springer-Verlag, Berlin-New York, 1974. Held at the University of Victoria, Victoria, B. C., 8–11 May 1972. MR**0472459****[15]**Peter A. Loeb,*Conversion from nonstandard to standard measure spaces and applications in probability theory*, Trans. Amer. Math. Soc.**211**(1975), 113–122. MR**0390154**, https://doi.org/10.1090/S0002-9947-1975-0390154-8**[16]**Peter A. Loeb,*Applications of nonstandard analysis to ideal boundaries in potential theory*, Israel J. Math.**25**(1976), no. 1-2, 154–187. MR**0457757**, https://doi.org/10.1007/BF02756567**[17]**W. A. J. Luxemburg,*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (Inte rnat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR**0244931****[18]**Moshé Machover and Joram Hirschfeld,*Lectures on non-standard analysis*, Lecture Notes in Mathematics, Vol. 94, Springer-Verlag, Berlin-New York, 1969. MR**0249285****[19]**R. Parikh and M. Parnes,*Conditional probability can be defined for all pairs of sets of reals*, Advances in Math.**9**(1972), 313–315. MR**0324736**, https://doi.org/10.1016/0001-8708(72)90022-9**[20]**Rohit Parikh and Milton Parnes,*Conditional probabilities and uniform sets*, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972) Springer, Berlin, 1974, pp. 180–194. Lecture Notes in Math., Vol. 369. MR**0482898****[21]**Salim Rashid,*Economies with infinitely many traders*, Ph.D. Dissertation, Yale Univ., New Haven, Ct., 1976.**[22]**Abraham Robinson,*Non-standard analysis*, North-Holland Publishing Co., Amsterdam, 1966. MR**0205854****[23]**Walter Rudin,*Real and complex analysis*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR**0344043****[24]**K. D. Stroyan and W. A. J. Luxemburg,*Introduction to the theory of infinitesimals*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 72. MR**0491163****[25]**Douglas N. Hoover and Edwin Perkins,*Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II*, Trans. Amer. Math. Soc.**275**(1983), no. 1, 1–36, 37–58. MR**678335**, https://doi.org/10.1090/S0002-9947-1983-0678335-1**[26]**Tom L. Lindstrøm,*Hyperfinite stochastic integration. I. The nonstandard theory*, Math. Scand.**46**(1980), no. 2, 265–292. MR**591606**, https://doi.org/10.7146/math.scand.a-11868

Tom L. Lindstrøm,*Hyperfinite stochastic integration. II. Comparison with the standard theory*, Math. Scand.**46**(1980), no. 2, 293–314. MR**591607**, https://doi.org/10.7146/math.scand.a-11869

Tom L. Lindstrøm,*Hyperfinite stochastic integration. III. Hyperfinite representations of standard martingales*, Math. Scand.**46**(1980), no. 2, 315–333. MR**591608**, https://doi.org/10.7146/math.scand.a-11870**[27]**Peter A. Loeb,*Weak limits of measures and the standard part map*, Proc. Amer. Math. Soc.**77**(1979), no. 1, 128–135. MR**539645**, https://doi.org/10.1090/S0002-9939-1979-0539645-6**[28]**Peter A. Loeb,*An introduction to nonstandard analysis and hyperfinite probability theory*, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 105–142. MR**556680****[29]**Frank Wattenberg,*Nonstandard measure theory: avoiding pathological sets*, Trans. Amer. Math. Soc.**250**(1979), 357–368. MR**530061**, https://doi.org/10.1090/S0002-9947-1979-0530061-4

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654856-1

Keywords:
Radon measures,
measure-preserving maps,
martingales,
uniform integrability,
compactifications,
nonstandard analysis

Article copyright:
© Copyright 1982
American Mathematical Society