Consistency of Brownian finite-dimensional laws

For $n=0$, set ${\mu }_{()}$ equal to the unique law on the singleton empty tuple. For $n\ge 1$, [F1] makes $\mathrm{\Sigma }(\mathbf{t})$ an admissible covariance matrix, so [F2] supplies the centered law ${\mu }_{\mathbf{t}}=N_n(0,\mathrm{\Sigma }(\mathbf{t}))\mathrm{.}$ This remains well-defined when a time is zero, times repeat, or the covariance is singular.

Let $\theta :\{1,\dots ,m\}\to \{1,\dots ,n\}$ be any map, let $R:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be the coordinate map $(Rx{)}_j=x_{\theta (j)}$, and take $X\sim {\mu }_{\mathbf{t}}$. For $u\in {\mathbb{R}}^m$, [F3] gives $E{e}^{iu\cdot RX}=E{e}^{i(R^{T}u)\cdot X}=\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{u}^{T}R\mathrm{\Sigma }(\mathbf{t})R^{T}u)\mathrm{.}$ The $(j,k)$ entry of $R\mathrm{\Sigma }(\mathbf{t})R^T$ is $\min (t_{\theta (j)},t_{\theta (k)})$, so [F3] identifies the law of $RX$ with ${\mu }_{(t_{\theta (1)},\dots ,t_{\theta (m)})}$. The case $m=0$ is the unique empty-tuple law.

A bijective $\theta$ permutes coordinates, an injective coordinate selection deletes the unselected coordinates, and a noninjective $\theta$ repeats coordinates. Thus step 2.1 proves all three promised compatibilities, with both orders of any inverse permutation covered. AC is spent only in the existence and characteristic-function uniqueness suppliers [F2]–[F3].