Universal Loeb-measurability of sets and of the standard part map with applications

Authors:
D. Landers and L. Rogge

Journal:
Trans. Amer. Math. Soc. **304** (1987), 229-243

MSC:
Primary 28E05; Secondary 03H05

DOI:
https://doi.org/10.1090/S0002-9947-1987-0906814-1

MathSciNet review:
906814

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown in this paper that for -saturated models many important external sets of nonstandard analysis--such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points--are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction.

Moreover, the standard part map can be used as a measure preserving transformation for all -smooth measures, and not only for Radon-measures as known up to now.

Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of -smooth Baire-measures to -smooth Borel-measures, the decomposition theorems for -smooth Baire-measures and -smooth Borel-measures and Kakutani's theorem for product measures.

**[1]**S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, and T. Lindstrøm,*Nonstandard methods in stochastic analysis and mathematical physics*, Academic Press, New York, 1986. MR**859372 (88f:03061)****[2]**Robert M. Anderson,*Star-finite representations of measure spaces*, Trans. Amer. Math. Soc.**271**(1982), 667-687. MR**654856 (83m:03077)****[3]**Robert M. Anderson and Salim Rashid,*A nonstandard characterization of weak convergence*, Proc. Amer. Math. Soc.**69**(1978), 327-332. MR**0480925 (58:1073)****[4]**Nigel J. Cutland,*Nonstandard measure theory and its applications*, Bull. London Math. Soc.**15**(1983), 525-589. MR**720746 (85b:28001)****[5]**C. Ward Henson,*Analytic sets, Baire sets and the standard part map*, Canad. J. Math.**31**(1979), 663-672. MR**536371 (80i:28019)****[6]**J. D. Knowles,*Measures on topological spaces*, Proc. London Math. Soc. (3)**17**(1967), 139-156. MR**0204602 (34:4441)****[7]**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 (52:10980)****[8]**-,*Weak limits of measures and the standard part map*, Proc. Amer. Math. Soc.**77**(1979), 128-135. MR**539645 (80i:28020)****[9]**-,*An introduction to nonstandard analysis and hyperfinite probability theory*, Probabilistic Analysis and Related Topics, vol. 2 (A. T. Bharucha-Reid, ed.), Academic Press, New York, 1979, pp. 105-142. MR**556677 (80j:60001)****[10]**-,*Measure spaces in nonstandard models underlying standard stochastic processes*, Proc. Internat. Congr. Math., Warzaw, 1983.**[11]**-,*A functional approach to nonstandard measure theory*, Conference on Modern Analysis and Probability (Beals et al., eds), Amer. Math. Soc., Providence, R.I., 1984. MR**737406 (86b:28026)****[12]**W. A. J. Luxemburg,*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (W. A. J. Luxemburg, ed.), Holt, Rinehart and Winston, New York, 1969, pp. 18-86. MR**0244931 (39:6244)****[13]**A. Saponakis and M. Sion,*On generation of Radon-like measures*, Lecture Notes in Math., vol. 1033, Springer-Verlag, Berlin and New York, 1983, pp. 283-294. MR**729544 (85g:28005)****[14]**K. D. Stroyan and W. A. J. Luxemburg,*Introduction to the theory of infinitesimals*, Academic Press, New York, 1976. MR**0491163 (58:10429)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28E05,
03H05

Retrieve articles in all journals with MSC: 28E05, 03H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0906814-1

Keywords:
Loeb-measures,
Baire- and Borel-measures,
representation and extension of measures,
-saturated models

Article copyright:
© Copyright 1987
American Mathematical Society