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

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