$d$-dimensional Brownian motion

We first record the diagonal-Gaussian fact used in both directions. If $X\sim N_d(0,h{I}_d)$ for $h\ge 0$, [F2] realizes that law as $\sqrt{h}Z$, where the coordinates of $Z$ are independent standard normals. Equality of vector laws preserves every rectangle probability, so the coordinates of $X$ are independent and each has law $N(0,h)$. This includes $h=0$, when every coordinate is the constant zero variable.

Conversely, suppose that $X_1,\dots ,X_d$ are independent and each has law $N(0,h)$. For $u\in {\mathbb{R}}^d$, [F3] gives $\mathbb{E}{e}^{iu\cdot X}={\prod }_{\alpha =1}^d\mathbb{E}{e}^{i{u}_{\alpha }{X}_{\alpha }}=\exp \text{\hspace{0.17em}⁣}(-\frac{h}{2}{\sum }_{\alpha =1}^d{u}_{\alpha }^2)\mathrm{.}$ This is the characteristic function of the existing law $N_d(0,h{I}_d)$ by [F2], and the uniqueness clause of [F2] identifies the vector laws. Thus $X\sim N_d(0,h{I}_d)$. For $d=1$ this says exactly that $N_1(0,h)$ is the scalar $N(0,h)$ law; for $h=0$ both sides are the point mass at zero.

Each ${\mathrm{\Phi }}_{\alpha }$ in the Definition is a random element: for every finite $F\subseteq I$, the finite observation map $\omega \mapsto (B_t^{\alpha }(\omega ){)}_{t\in F}$ is measurable because the inverse image of each coordinate generator is measurable. Therefore the inverse images of all finite-coordinate cylinders, and hence of their generated cylinder sigma-algebra, are measurable.

Assume first that $B$ satisfies the three vector clauses of the Definition. Fix $0=t_0<\cdots <t_n$ and put $Z_j=B_{t_j}-B_{t_{j-1}}$. The random vectors $Z_j$ are independent, and by step 1.1 the coordinates $(Z_j^{\alpha }{)}_{\alpha =1}^d$ within each fixed $j$ are independent $N(0,t_j-t_{j-1})$ variables. Hence, for arbitrary Borel sets $A_{j,\alpha }\subseteq \mathbb{R}$, $\begin{array}{rl}{\displaystyle P\text{\hspace{0.17em}⁣}(\bigcap _{j=1}^n\bigcap _{\alpha =1}^d\{Z_j^{\alpha }\in A_{j,\alpha }\})}& {\displaystyle =\prod _{j=1}^{n}P\text{\hspace{0.17em}⁣}(Z_j\in \prod _{\alpha =1}^d{A}_{j,\alpha })}\\ {\displaystyle }& {\displaystyle =\prod _{j=1}^n\prod _{\alpha =1}^{d}P(Z_j^{\alpha }\in A_{j,\alpha })\mathrm{.}}\end{array}$ The rectangle criterion [F4] therefore makes the whole finite family $(Z_j^{\alpha }{)}_{j,\alpha }$ mutually independent.

For a fixed $\alpha$, step 2.1 gives independent increments with the required scalar normal laws. The vector condition at time zero gives $B_0^{\alpha }=0$ almost surely. On the vector continuity event every coordinate path is continuous by [F6]. Consequently each $B^{\alpha }$ is standard one-dimensional Brownian motion by [F1].

For each $\alpha$, let ${\mathrm{\Pi }}_{\alpha }$ be the pullbacks by ${\mathrm{\Phi }}_{\alpha }$ of finite-coordinate cylinders. These classes are pi-systems containing $\mathrm{\Omega }$ by [F5], and $\sigma ({\mathrm{\Pi }}_{\alpha })=\sigma ({\mathrm{\Phi }}_{\alpha })$. To check their independence, choose one cylinder from every member of any finite subfamily of the coordinates and take the sorted union of their finitely many time supports, adjoining time zero. On the probability-one event $\{B_0=0\}$, each coordinate's finite observation vector equals a measurable affine function of its own block of scalar increments. Thus its cylinder probabilities equal those of that affine function. Step 2.1 makes all scalar increments independent; [F4] first groups them by $\alpha$ and then preserves independence under these measurable affine maps. The chosen cylinder probabilities therefore factor. Thus the pi-systems ${\mathrm{\Pi }}_{\alpha }$ are independent, and [F5] makes the full random elements ${\mathrm{\Phi }}_1,\dots ,{\mathrm{\Phi }}_d$ independent. This proves the forward implication. Empty-support cylinders give $\mathrm{\Omega }$ or $\mathrm{\varnothing }$ and obey the same formula.

Conversely, assume that the random elements ${\mathrm{\Phi }}_1,\dots ,{\mathrm{\Phi }}_d$ are independent and that every $B^{\alpha }$ is a standard one-dimensional Brownian motion. Fix $0=t_0<\cdots <t_n$ and let $U_{\alpha }=(B_{t_j}^{\alpha }-B_{t_{j-1}}^{\alpha }{)}_{j=1}^n\mathrm{.}$ This is a measurable function of ${\mathrm{\Phi }}_{\alpha }$, so [F4] makes $U_1,\dots ,U_d$ independent. Within each $U_{\alpha }$, the scalar Brownian increments are independent and the $j$th has law $N(0,t_j-t_{j-1})$ by [F1]. For arbitrary Borel $A_{j,\alpha }$, the same two-stage rectangle calculation as in step 2.1, now first over $\alpha$ and then over $j$, gives $P\text{\hspace{0.17em}⁣}({\bigcap }_{\alpha =1}^d{\bigcap }_{j=1}^n\{U_{\alpha ,j}\in A_{j,\alpha }\})={\prod }_{\alpha =1}^d{\prod }_{j=1}^{n}P(U_{\alpha ,j}\in A_{j,\alpha })\mathrm{.}$ Hence all the scalar variables $U_{\alpha ,j}$ are mutually independent.

Group the independent scalar variables of step 3.3 by their time index $j$. By [F4], the resulting sigma-algebras are independent. Each vector increment $Z_j=(U_{1,j},\dots ,U_{d,j})=B_{t_j}-B_{t_{j-1}}$ is measurable with respect to the $j$th grouped sigma-algebra, so the vectors $Z_1,\dots ,Z_n$ are independent. For fixed $j$, its coordinates are independent $N(0,t_j-t_{j-1})$ variables; step 1.2 therefore gives $Z_j\sim N_d(0,(t_j-t_{j-1})I_d)$.

For every $\alpha$, let $C_{\alpha }$ be a probability-one event on which the path $B^{\alpha }$ is continuous, and let $E_{\alpha }=\{B_0^{\alpha }=0\}$. Because $d$ is finite, repeated finite subadditivity in [F6] gives $P\text{\hspace{0.17em}⁣}({\bigcap }_{\alpha =1}^d(C_{\alpha }\cap E_{\alpha }))=1.$ On this one event, $B_0=0$ and the vector path is continuous by the coordinatewise criterion [F6]. Together with step 4.1 this proves the three vector clauses, and hence the reverse implication.

The empty increment list is vacuous; a one-increment list is covered by steps 2.1 and 4.1; repeated times and zero-length increments are excluded by the strictly increasing convention, while time zero and the singular zero-variance law are handled in steps 1.1, 1.2, 3.1, and 5.1. The assumption $d\ge 1$ excludes the empty-coordinate process. AC is used through [F1]--[F3] for the normal-law and Brownian interfaces and their law uniqueness; the finite regrouping, cylinder, and continuity arguments add no further choice.