Kolmogorov construction of the canonical Gaussian process

By [F5], the usual real line is complete and has the countable dense subset $\mathbb{Q}$, hence is Polish; its Borel measurable space is therefore standard Borel via the identity presentation. This verifies, rather than merely assumes, the coordinate-space hypothesis of [F2].

For every finite $F\subseteq [0,\mathrm{\infty })$, use [F1] to put on ${\mathbb{R}}^F$ the centered Gaussian law with covariance $(\min (s,t){)}_{s,t\in F}$; step 1.1 and [F2] give a probability $P$ on the cylinder sigma-algebra of ${\mathbb{R}}^{[0,\mathrm{\infty })}$ with exactly those marginals. The empty marginal has mass one on the singleton empty product.

Let $X_t(x)=x(t)$. By [F3], every finite evaluation vector of $X$ has the prescribed centered Gaussian law. Thus [F4] makes $X$ Gaussian, and its one- and two-coordinate marginals give $E[X_t]=0$ and $E[X_s{X}_t]=\min (s,t)$, including $X_0=0$ almost surely because its variance is zero.

The conclusion of [F2] is only a measure on the cylinder sigma-algebra realizing finite-coordinate laws; neither [F2] nor [F3] supplies a common full-measure set on which $t\mapsto X_t(x)$ is continuous. Consequently step 3.1 does not establish the path-continuity clause needed for Brownian motion. AC is used in [F1]–[F2] and the Gaussian interface [F4], including the choices inside arbitrary-index Kolmogorov extension; no stronger regularity is inferred from it.