I'm looking for a "canonical" probability measure $\nu$ on the space $C^0([0,1];\mathbb{R})$, endowed with the Borel $\sigma$-algebra.
I know I can take any measurable function $f:C^0([0,1];\mathbb{R})\to C^0([0,1];\mathbb{R})$ and define $\nu:=f_\#\mu$, the push-forward of the measure $\mu$ of the Wiener Process, however, it doesn't look quite satisfactory to me: the gaussianity of the measure is really good, but covariances are rather asymmetrical.
I was thinking about defining it through the marginals $\nu(\pi^{-1}_{t_1}(A_1)\cap\dots\cap\pi^{-1}_{t_k}(A_k))=\mathcal N(A_1)\times\cdots\times\mathcal N(A_k),$ where $\mathcal N$ is the standard Gaussian measure.
However, I think it's not possible to apply Kolmogorov Continuity Theorem for it, making it a measure on $\mathbb{R}^{[0,1]}$ and not on the space of continuous functions: maybe it is a white noise?
Is there some canonical construction that I maybe do not know but my be useful to me?