Wiener measure on continuous path space

By [F1], fix $B$ and $A$ as in the Definition. Every path $W(\omega )$ is continuous: on $A$ it is a Brownian path, and off $A$ it is the zero path. For fixed $t$, ${\tilde{B}}_t$ is measurable because for each Borel $E$, its inverse image is $(A\cap \{B_t\in E\})$ together with $A^c$ exactly when $0\in E$.

Fix $f\in C([0,\mathrm{\infty }),\mathbb{R})$ and $n\ge 1$. By path continuity and density [F4], $M_{n,f}(\omega ):={\max }_{0\le t\le n}\mathrm{\mid }W(\omega )(t)-f(t)\mathrm{\mid }={\sup }_{q\in \mathbb{Q}\cap [0,n]}\mathrm{\mid }{\tilde{B}}_q(\omega )-f(q)\mathrm{\mid }\mathrm{.}$ Enumerate the countable rational set once. Each function under the supremum is measurable by step 1.1 and [F3], so [F3] makes $M_{n,f}$ measurable and finite.

By [F3], every finite partial sum $D_{N,f}={\sum }_{n=1}^N{2}^{-n}(1\wedge M_{n,f})$ is measurable; here $1\wedge M=1-(1-M{)}^{+}$. The partial sums converge pointwise to $d_{\mathrm{u}\mathrm{o}\mathrm{c}}(W,f)$ by [F2], so [F3] makes that distance measurable. Therefore the inverse image under $W$ of every open metric ball is measurable. By separability in [F2], fix a countable dense $D$; the balls with centers in $D$ and positive rational radii form a countable basis, by the metric triangle inequality and rational density, and [F5] makes every subfamily countable. Every open set is therefore a countable union of such balls. Since open sets generate the Borel sigma-algebra by [F5], $W$ is Borel measurable and hence a random element by [F6].

By [F6], $\mathsf{W}=P\circ W^{-1}$ is a probability measure. For any finite times $t_1,\dots ,t_k$, the vectors $({\tilde{B}}_{t_j}{)}_{j=1}^k$ and $(B_{t_j}{)}_{j=1}^k$ agree on $A$, hence almost surely, so their laws coincide. The coordinate vector $({\pi }_{t_j}{)}_{j=1}^k$ under $\mathsf{W}$ has exactly the former law by the pushforward definition. Thus the coordinate process under Wiener measure has every Brownian finite-dimensional law. Empty tuples have the unit law and ${\pi }_0=0$ almost surely.

The zero path used on $A^c$ is fixed and canonical. AC is used only through [F1] to obtain the normal-law construction and Brownian process; redefining a given process on its one supplied null event, taking fixed rational suprema, and pushing forward use no further choice.