 Wiener theorem

There exists a probability measure on $C_0([0,\infty ),\mathbb{R})$ under which the coordinate process is standard Brownian motion, and its distribution there is unique. Construct consistent centered Gaussian finite-dimensional laws with covariance $\min (s,t)$ and use the Kolmogorov extension theorem. The fourth increment moment $3|t-s{|}^2$ supplies a continuous modification by the Kolmogorov continuity theorem. Its independent increments and normal increment distributions are unchanged. Uniqueness follows because coordinate evaluations at rational times generate the Borel sigma-algebra of continuous-path space.