On a measure in Wiener space and applications
HTML articles powered by AMS MathViewer
- by K. S. Ryu and M. K. Im PDF
- Trans. Amer. Math. Soc. 355 (2003), 2205-2222 Request permission
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.References
- M. D. Brue, A functional transform for Feynman integral similar to the Fourier transform, Ph. D. Dissertation, U. Minnesota (1972).
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
- R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1–30. MR 404571, DOI 10.1307/mmj/1029001617
- K. S. Chang and K. S. Ryu, A generalized converse measurability theorem, Proc. Amer. Math. Soc. 104 (1988), no. 3, 835–839. MR 935104, DOI 10.1090/S0002-9939-1988-0935104-2
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
- 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
- G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157–176. MR 555044, DOI 10.2140/pjm.1979.83.157
- G. W. Johnson and D. L. Skoug, An $L_{p}$ analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103–127. MR 514964, DOI 10.1307/mmj/1029002166
- E. J. McShane, Families of measures and representations of algebras of operators, Trans. Amer. Math. Soc. 102 (1962), 328–345. MR 137002, DOI 10.1090/S0002-9947-1962-0137002-X
- M. M. Rao, Measure theory and integration, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 891879
- H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
- K. S. Ryu, A property of Borel subsets of Wiener space, J. Chungcheng Math. Soc., Vol. 14 (1991), 45-48.
- N. Wiener, Differential space, J. Math. Phys., 58 (1923), 131-174.
- Y. Yamasaki, Measures on infinite-dimensional spaces, Series in Pure Mathematics, vol. 5, World Scientific Publishing Co., Singapore, 1985. MR 999137, DOI 10.1142/0162
- J. Yeh, Stochastic processes and the Wiener integral, Pure and Applied Mathematics, Vol. 13, Marcel Dekker, Inc., New York, 1973. MR 0474528
Additional Information
- K. S. Ryu
- Affiliation: Department of Mathematics, Han Nam University, Taejon 306-791, Korea
- Email: ksr@math.hannam.ac.kr
- M. K. Im
- Affiliation: Department of Mathematics, Han Nam University, Taejon 306-791, Korea
- Email: mki@mail.hannam.ac.kr
- 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. 2001-1-10100-011-1 from the Basic Research Program of the Korea Science $\&$ Engineering Foundation.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2205-2222
- MSC (2000): Primary 28C20, 44A15, 46G12, 46T12, 58D20
- DOI: https://doi.org/10.1090/S0002-9947-03-03190-8
- MathSciNet review: 1973988