On a measure in Wiener space and applications
Authors:
K. S. Ryu and M. K. Im
Journal:
Trans. Amer. Math. Soc. 355 (2003), 22052222
MSC (2000):
Primary 28C20, 44A15, 46G12, 46T12, 58D20
Published electronically:
February 4, 2003
MathSciNet review:
1973988
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space.
 1.
M. D. Brue, A functional transform for Feynman integral similar to the Fourier transform, Ph. D. Dissertation, U. Minnesota (1972).
 2.
R.
H. Cameron, The translation pathology of Wiener space, Duke
Math. J. 21 (1954), 623–627. MR 0065033
(16,375b)
 3.
R.
H. Cameron and W.
T. Martin, The behavior of measure and
measurability under change of scale in Wiener space, Bull. Amer. Math. Soc. 53 (1947), 130–137. MR 0019259
(8,392a), http://dx.doi.org/10.1090/S000299041947087620
 4.
R.
H. Cameron and D.
A. Storvick, An 𝐿₂ analytic FourierFeynman
transform, Michigan Math. J. 23 (1976), no. 1,
1–30. MR
0404571 (53 #8371)
 5.
K.
S. Chang and K.
S. Ryu, A generalized converse measurability
theorem, Proc. Amer. Math. Soc.
104 (1988), no. 3,
835–839. MR
935104 (89e:28021), http://dx.doi.org/10.1090/S00029939198809351042
 6.
Edwin
Hewitt and Karl
Stromberg, Real and abstract analysis. A modern treatment of the
theory of functions of a real variable, SpringerVerlag, New York,
1965. MR
0188387 (32 #5826)
 7.
Gerald
W. Johnson and Michel
L. Lapidus, The Feynman integral and Feynman’s operational
calculus, Oxford Mathematical Monographs, The Clarendon Press, Oxford
University Press, New York, 2000. Oxford Science Publications. MR 1771173
(2001i:58015)
 8.
G.
W. Johnson and D.
L. Skoug, Scaleinvariant measurability in Wiener space,
Pacific J. Math. 83 (1979), no. 1, 157–176. MR 555044
(81b:28016)
 9.
G.
W. Johnson and D.
L. Skoug, An 𝐿_{𝑝} analytic FourierFeynman
transform, Michigan Math. J. 26 (1979), no. 1,
103–127. MR
514964 (81a:46050)
 10.
E.
J. McShane, Families of measures and
representations of algebras of operators, Trans. Amer. Math. Soc. 102 (1962), 328–345. MR 0137002
(25 #462), http://dx.doi.org/10.1090/S0002994719620137002X
 11.
M.
M. Rao, Measure theory and integration, Pure and Applied
Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A
WileyInterscience Publication. MR 891879
(89k:28001)
 12.
H.
L. Royden, Real analysis, 3rd ed., Macmillan Publishing
Company, New York, 1988. MR 1013117
(90g:00004)
 13.
K. S. Ryu, A property of Borel subsets of Wiener space, J. Chungcheng Math. Soc., Vol. 14 (1991), 4548.
 14.
N. Wiener, Differential space, J. Math. Phys., 58 (1923), 131174.
 15.
Y.
Yamasaki, Measures on infinitedimensional spaces, Series in
Pure Mathematics, vol. 5, World Scientific Publishing Co., Singapore,
1985. MR
999137 (90b:28015)
 16.
J.
Yeh, Stochastic processes and the Wiener integral, Marcel
Dekker,#Inc., New York, 1973. Pure and Applied Mathematics, Vol. 13. MR 0474528
(57 #14166)
 1.
 M. D. Brue, A functional transform for Feynman integral similar to the Fourier transform, Ph. D. Dissertation, U. Minnesota (1972).
 2.
 R. H. Cameron, The translation pathology of Wiener space, Duke Math. J., 21 (1954), 623627. MR 16:375b
 3.
 R. H. Cameron and W. T. Martin, The behavior of measure and measurability under change of scale in Wiener space, Bull. Amer. Math. Soc., Vol. 53 (1947), 130137. MR 8:392a
 4.
 R. H. Cameron and D. A. Storvick, An analytic FourierFeynman transform, Michigan Math. J., 23 (1976), 130. MR 53:8371
 5.
 K. S. Chang and K. S. Ryu, A generalized converse measurability theorem, Proceedings of the American Mathematical Society, Vol. 104, No. 3 (1988), 835839. MR 89e:28021
 6.
 E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, SpringerVerlag, New York, (1965). MR 32:5826
 7.
 G. W. Johnson and M. L. Lapidus, The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). MR 2001i:58015
 8.
 G. W. Johnson and D. L. Skoug, Scaleinvariant measurability in Wiener space, Pacific J. Math., Vol. 83, No. 1 (1979), 157176. MR 81b:28016
 9.
 G. W. Johnson and D. L. Skoug, An analytic FourierFeynman transform, Michigan Math. J., 26 (1979), 103127. MR 81a:46050
 10.
 E. J. McShane, Families of measures and representations of algebras of operators, Trans. Amer. Math. Soc. 102 (1962), 328345. MR 25:462
 11.
 M. M. Rao, Measure theory and integration, Pure and Applied Mathematics, John Wiley and Sons Inc., New York (1987). MR 89k:28001
 12.
 H. L. Royden, Real analysis, Third edition, Macmillan Publishing Company, New York (1988). MR 90g:00004
 13.
 K. S. Ryu, A property of Borel subsets of Wiener space, J. Chungcheng Math. Soc., Vol. 14 (1991), 4548.
 14.
 N. Wiener, Differential space, J. Math. Phys., 58 (1923), 131174.
 15.
 Y. Yamasaki, Measures on infinitedimensional spaces, World Scientific Series in Pure Mathematics, Vol. 15 (1985). MR 90b:28015
 16.
 J. Yeh, Stochastic process and the Wiener integral, Pure and Applied Mathematics, Vol. 13, Marcel Dekker, Inc., New York (1973). MR 57:14166
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Additional Information
K. S. Ryu
Affiliation:
Department of Mathematics, Han Nam University, Taejon 306791, Korea
Email:
ksr@math.hannam.ac.kr
M. K. Im
Affiliation:
Department of Mathematics, Han Nam University, Taejon 306791, Korea
Email:
mki@mail.hannam.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002994703031908
PII:
S 00029947(03)031908
Keywords:
Wiener measure,
scaleinvariant measurability,
FourierFeynman transform
Received by editor(s):
April 6, 2001
Received by editor(s) in revised form:
August 29, 2002
Published electronically:
February 4, 2003
Additional Notes:
This work was supported by grant No. 20011101000111 from the Basic Research Program of the Korea Science $&$ Engineering Foundation.
Article copyright:
© Copyright 2003
American Mathematical Society
