Existence of continuous Brownian motion

By [F1], on the canonical coordinate probability space there is a centered Gaussian process $X$ with covariance $\min (s,t)$. In particular $X_0=0$ almost surely, and [F4] gives $X_t-X_s\sim N(0,\mathrm{\mid }t-s\mathrm{\mid })$ for all $s,t\ge 0$.

By [F2] with $m=2$, $E\mathrm{\mid }{X}_t-X_s{\mathrm{\mid }}^4=3\mathrm{\mid }t-s{\mathrm{\mid }}^2\mathrm{.}$ By [F6], the real target is complete and separable. Thus [F3], with $\alpha =4$, $\beta =1$, and the supplied constants $C_T=3$, gives a modification $B$ of $X$ and one probability-one event on which every path of $B$ is continuous.

Fix a finite time list $t_1,\dots ,t_n$. Since $B$ is a modification, each event $\{B_{t_j}\mathrm{\ne }{X}_{t_j}\}$ is null; [F7] makes their finite union null. Hence the two evaluation vectors agree almost surely and have the same law. This includes the empty list, for which both laws are the unit mass on the empty tuple. Consequently all finite-dimensional laws of $B$ equal those of $X$, so $B$ is centered Gaussian with covariance $\min (s,t)$ and $B_0=0$ almost surely.

Apply [F4] to $B$: it has the independent $N(0,t_j-t_{j-1})$ increments on every finite increasing list. Together with $B_0=0$ from step 3.1 and the common continuity event from step 2.1, [F5] says that $B$ is standard Brownian motion. AC is used through [F1], [F2], [F4], and [F5] for normal-law construction and the arbitrary-index extension; the continuity construction [F3] adds no choice.