On a measure in Wiener space and applications

Ryu, K.; Im, M.

doi:10.1090/S0002-9947-03-03190-8

On a measure in Wiener space and applications
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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