Remark

— For every semi-open interval $J = ]a, b]$ contained in T, let us set $l(J) = b - a$ (the length of J) and denote by $A_J$ the linear form $f \mapsto f(b) - f(a)$ on $\mathscr{C}$. It can be shown that the Wiener measure is characterized by the following property:

Let $J_1, \ldots, J_n$ be semi-open intervals contained in T and pairwise disjoint. The image of the measure w under the linear mapping $f \mapsto (A_{J_1}(f), \ldots, A_{J_n}(f))$ of $\mathscr{C}$ into $\mathbf{R}^n$ is equal to $\gamma_{a_1} \otimes \cdots \otimes \gamma_{a_n}$ with $a_i = l(J_i)^{1/2}$ for $1 \leq i \leq n$.