Optimal estimation of shell thickness in Cutland’s construction of Wiener measure
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- by Bang-He Li and Ya-Qing Li PDF
- Proc. Amer. Math. Soc. 126 (1998), 225-229 Request permission
Abstract:
In Cutland’s construction of Wiener measure, he used the product of Gaussian measures on $^*R^N$, where $N$ is an infinite integer. It is mentioned by Cutland and Ng that for the product measure $\gamma$, \[ \gamma (\{x:R_1\le \|x\|\le R_2\})\simeq 1,\] where $R_1=1-(\log N)^{\frac 12} N^{-\frac 12}$ and $R_2=1+MN^{-\frac 12}$ with $M$ any positive infinite number. We prove here that $R_1$ may be replaced by $1-mN^{-\frac 12}$ with $m$ any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that $\gamma (\{x:\|x\|<1 \})\simeq \gamma (\{x:\|x\|>1\})\simeq \frac 12$. And for the *Lebesgue measure $\mu$, $\mu (\{x:\|x\|\le r\})$ is finite and not infinitesimal iff $r=(2\pi e)^{-\frac 12}N^{\frac 12(1+\frac 1N)}e^{\frac aN}$ with $a$ finite, while for the *Lebesgue area of the sphere $S^{N-1}(r)$, $r$ should be $(2\pi e)^{-\frac 12}N^{\frac 12} e^{\frac aN}$.References
- Nigel J. Cutland, Infinitesimals in action, J. London Math. Soc. (2) 35 (1987), no. 2, 202–216. MR 881511, DOI 10.1112/jlms/s2-35.2.202
- Nigel Cutland and Siu-Ah Ng, The Wiener sphere and Wiener measure, Ann. Probab. 21 (1993), no. 1, 1–13. MR 1207212
- Bang He Li and Ji Jiang Zhang, On the Dedekind completion of ${}^*\textbf {R}$, Systems Sci. Math. Sci. 1 (1988), no. 1, 29–39. MR 1009766
Additional Information
- Bang-He Li
- Affiliation: Institute of Systems Science, Academia Sinica, Beijing 100080, People’s Republic of China
- Email: libh@iss06.iss.ac.cn
- Ya-Qing Li
- Affiliation: Institute of Systems Science, Academia Sinica, Beijing 100080, People’s Republic of China
- Email: yli@iss06.iss.ac.cn
- Received by editor(s): July 14, 1995
- Received by editor(s) in revised form: April 9, 1996
- Additional Notes: This project was supported by the National Natural Science Foundation of China.
- Communicated by: Andreas R. Blass
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 225-229
- MSC (1991): Primary 03H05, 28E05, 51M05; Secondary 28A35, 28C20
- DOI: https://doi.org/10.1090/S0002-9939-98-03888-X
- MathSciNet review: 1396985