= Wiener theorem
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= Wiener's theorem
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{synonym}
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.
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