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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Optimal estimation of shell thickness
in Cutland's construction of Wiener measure


Authors: Bang-He Li and Ya-Qing Li
Journal: Proc. Amer. Math. Soc. 126 (1998), 225-229
MSC (1991): Primary 03H05, 28E05, 51M05; Secondary 28A35, 28C20
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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 $\gamma$,

\begin{displaymath}\gamma(\{x:R_1\le \|x\|\le R_2\})\simeq 1,\end{displaymath}

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 $(2\pi e)^{-\frac 12}N^{\frac 12} e^{\frac aN}$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-98-03888-X
PII: S 0002-9939(98)03888-X
Keywords: Shell thickness, Wiener measure, $*$-finite Euclidean space
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
Article copyright: © Copyright 1998 American Mathematical Society