Uniqueness of Wiener measure

By [F1], Wiener measure is a probability measure on $\mathcal{B}(C)$ and its coordinate process is centered Gaussian with covariance $\min (s,t)$. Thus at least one probability with the stated property exists. AC is used here through the normal-law and Brownian construction interfaces; the uniqueness argument below makes no further use of it.

Let $\mu$ and $\nu$ be two probabilities with the stated property. By [F2], for every $k\ge 1$, every nonnegative rational list $q_1,\dots ,q_k$ (with repetitions allowed), and Borel sets $A_i\subseteq \mathbb{R}$, $\mu {\bigcap }_{i=1}^k\{f:f(q_i)\in A_i\}=\nu {\bigcap }_{i=1}^k\{f:f(q_i)\in A_i\}\mathrm{.}$ Indeed this is equality of the two finite-dimensional laws on the Borel rectangle $A_1\times \cdots \times A_k$. The same equality is trivial for the empty intersection $C$.

Let $\mathcal{P}$ be the family of the finite intersections in step 1.2, including the empty intersection $C$. It is a pi-system because the intersection of two members is obtained by concatenating their two finite lists. Moreover $\sigma (\mathcal{P})=\mathcal{B}(C)$: the family contains every one-coordinate set ${\pi }_q^{-1}(A)$, while every member is a finite intersection of such sets; now apply [F3].

Put $\mathcal{D}=\{A\in \mathcal{B}(C):\mu (A)=\nu (A)\}$. It contains $C$ because both measures have total mass one. If $A\subseteq B$ and $A,B\in \mathcal{D}$, finite additivity gives $\mu (B\setminus A)=\mu (B)-\mu (A)=\nu (B)-\nu (A)=\nu (B\setminus A)$. If $A_n\uparrow A$ with every $A_n\in \mathcal{D}$, continuity from below gives $\mu (A)={\lim }_n\mu (A_n)={\lim }_n\nu (A_n)=\nu (A)$. Hence $\mathcal{D}$ is a lambda-system. Step 1.2 says $\mathcal{P}\subseteq \mathcal{D}$, so [F4] and step 2.1 give $\mathcal{B}(C)=\sigma (\mathcal{P})\subseteq \mathcal{D}$. Thus $\mu =\nu$.

Step 1.1 proves existence and step 3.1 proves uniqueness, so Wiener measure is exactly the asserted probability. The covariance at time zero is zero, singleton laws and singular repeated-time vectors are covered by [F2], and both directions of “exactly one” have been established.