Conversion from nonstandard to standard measure spaces and applications in probability theory
Author:
Peter A. Loeb
Journal:
Trans. Amer. Math. Soc. 211 (1975), 113122
MSC:
Primary 28A10; Secondary 02H25, 60J99
MathSciNet review:
0390154
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard algebra containing the algebra , where the extended realvalued measure on is generated by the standard part of . If f is measurable, then its standard part is measurable on X. If f and are finite, then the integral of f is infinitely close to the integral of . Applications include coin tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the jth event and a construction of standard sample functions for Poisson processes.
 [1]
Allen
R. Bernstein and Peter
A. Loeb, A nonstandard 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
(58 #11313)
 [2]
William
Feller, An introduction to probability theory and its applications.
Vol. I, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd.,
London, 1957. 2nd ed. MR 0088081
(19,466a)
 [3]
Paul
G. Hoel, Sidney
C. Port, and Charles
J. Stone, Introduction to probability theory, Houghton Mifflin
Co., Boston, Mass., 1971. The Houghton Mifflin Series in Statistics. MR 0358880
(50 #11337c)
 [4]
Peter
A. Loeb, A nonstandard representation of measurable spaces,
𝐿_{∞}, and 𝐿*_{∞}, Contributions to
nonstandard analysis (Sympos., Oberwolfach, 1970), NorthHolland,
Amsterdam, 1972, pp. 65–80. Studies in Logic and Found. Math.,
Vol. 69. MR
0482128 (58 #2215)
 [5]
Peter
A. Loeb, A nonstandard representation of Borel measures and
𝜎finite measures, Victoria Symposium on Nonstandard Analysis
(Univ. Victoria, Victoria, B.C., 1972) Springer, Berlin, 1974,
pp. 144–152. Lecture Notes in Math., Vol. 369. MR 0476992
(57 #16537)
 [6]
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
(39 #6244)
 [7]
Abraham
Robinson, Nonstandard analysis, NorthHolland Publishing Co.,
Amsterdam, 1966. MR 0205854
(34 #5680)
 [8]
H.
L. Royden, Real analysis, The Macmillan Co., New York;
CollierMacmillan Ltd., London, 1963. MR 0151555
(27 #1540)
 [1]
 A. Bernstein and P. A. Loeb, A nonstandard integration theory for unbounded functions, Victoria Sympos. on Nonstandard Analysis, Lecture Notes in Math., vol. 369, SpringerVerlag, New York, 1974, pp. 4049. MR 0492167 (58:11313)
 [2]
 W. K. Feller, An introduction to probability theory and its applications. Vol. I, 2nd ed., Wiley, New York; Chapman and Hall, London, 1957. MR 19, 466. MR 0088081 (19:466a)
 [3]
 P. G. Hoel, S. Port and C. Stone, Introduction to probability theory, Houghton Mifflin, Boston, Mass., 1971. MR 0358880 (50:11337c)
 [4]
 P. A. Loeb, A nonstandard representation of measurable spaces, and , Contributions to Nonstandard Analysis, edited by W. A. J. Luxemburg and A. Robinson, NorthHolland, Amsterdam, 1972, pp. 6580. MR 0482128 (58:2215)
 [5]
 , A nonstandard representation of Borel measures and finite measures, Victoria Sympos. on Nonstandard Analysis, Lecture Notes in Math., vol 369, Springer Verlag, New York, 1974, pp. 144152. MR 0476992 (57:16537)
 [6]
 W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis and Probability (Internat. Sympos., Pasadena, Calif., 1967), Holt, Rinehart and Winston, New York, 1969, pp. 1886. MR 39 #6244. MR 0244931 (39:6244)
 [7]
 A. Robinson, Nonstandard analysis, NorthHolland, Amsterdam, 1966. MR 34 #5680. MR 0205854 (34:5680)
 [8]
 H. L. Royden, Real analysis, Macmillan, New York, 1968. MR 0151555 (27:1540)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
28A10,
02H25,
60J99
Retrieve articles in all journals
with MSC:
28A10,
02H25,
60J99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197503901548
PII:
S 00029947(1975)03901548
Keywords:
Measure space,
probability space,
standard integral,
coin tossing,
Poisson processes,
sample function,
strong Markov property
Article copyright:
© Copyright 1975
American Mathematical Society
