Let $p(t,x)$ denote the Gaussian density on $\mathbb{R}^d$ with variance $t$. To each finite partition $D=\{0=t_0,\dots,t_n=1\}$ of $[0,1]$ associate the measure on $(R^d)^n$, $\mu_D$ by $$\mu_D(dx)=\Pi_{i=1}^np(t_{i+1}-t_i, x_i-x_{i-1})dx_i.$$ These are the transition densities of $d$-dimensional BM started at $0$ and can be viewed as measures on path space - the product space $(R^d)^{[0,1]}$. Kolmogorov extension theorem then tells us that this net of measures $\mu_D$ extend to a measure $\mu$ on $(R^d)^{[0,1]}$, known as Wiener measure (the law of BM) -- once you fiddle around with Kolmogorov continuity theorem.
Now my question is, in what sense can do we have $\lim_{|D|\to 0}\mu_D=\mu$? In particular, for which path-functionals $f:(R^d)^{[0,1]}\to R$ do we have that $\mu_D(f)\to\mu(f)$? Even vague ideas/references would be much appreciated!