Families of Consistent Probability Measures

This paper deals with the following problem. Suppose that $(P_t)_{t\ge 0}$ is a family of consistent probability measures defined on a filtration $(\mathscr{F}_t)_{t\ge 0}$. Does there exist a measure $P$ on the $\sigma$-field $\vee_{t\geq 0}\mathscr{F}_t$ such that $P\,|\,\mathscr{F}_t=P_t$? The answer is positive for the spaces $C(\mathbf{R}_+,\mathbf{R}^d)$ and $D(\mathbf{R}_+,\mathbf{R}^d)$ endowed with the natural filtration. We prove this statement using a simple method based on the Prokhorov criterion of weak compactness.