Kolmogorov extension theorem for measures on the space of continuous functions

Question

In a text I am reading occurs the following:

One has a family of probability measures $(P_T)_{T\geq 0}$ such that $P_T$ is a measure on the measurable space $(C([0,T],\Bbb R^d), \mathcal B (C([0,T],\Bbb R^d)) )$.

It is stated that consistency of the family $(P_T)_{T\geq 0}$ is yielding a measure $P$ on $C([0,\infty),\Bbb R^d)$.

1. I think consistency means if we take $\pi^S_T: C([0,S],\Bbb R^d) \to C([0,T],\Bbb R^d)$,$f\mapsto f\vert_{[0,T]}$, then for every $S\geq T$ we have $P_T = P_S \circ (\pi_T^S)^{-1}$.

2. P seems to satisfy $P_T = P\circ(\pi_T)^{-1}$, where $\pi_T: C([0,\infty),\Bbb R^d) \to C([0,T],\Bbb R^d)$,$f\mapsto f\vert_{[0,T]}$

How is this done? My first thought was to apply the Kolmogorov extension theorem but I did not come far enough with it.

Answer 1

I apologize for this being so late, but I have a proposal for a solution.

I believe you can implement an outer measure construction of the desired $\mathrm P: \mathcal B(C([0,\infty), \mathbb R^d)) \rightarrow [0,1]$. Define a premeasure $\mathrm P_\infty$ on the ring generated by the closed balls in $C([0,\infty), \mathbb R^d)$, induced by the following definition on said balls. $$\mathrm P_\infty\big( C_\delta(x) \big) = \lim_{n\rightarrow\infty} \mathrm P_n\big( C_\delta(\pi_n(x)) \big)$$ where $\pi_T: C([0,\infty),\mathbb R^d) \rightarrow C([0,T], \mathbb R^d)$ is the restriction map $f \mapsto f|_{[0,T]}$ for each $T > 0$ and $C_\delta(y)$ is the closed ball of radius $\delta$ around $y$ in the relevant topological space.

This intuition comes from the fact that, for any $x \in C([0,\infty),\mathbb R^d)$ and $\delta > 0$, $$ C_\delta(x) = \bigcap_{n=1}^\infty \pi_n^{-1}(C_\delta(\pi_n(x))), $$ If the measure $\mathrm P$ in fact existed, the decreasing nature of the sets $\pi_n^{-1}(C_\delta(\pi_n(x)))$ would produce $$ \mathrm P\big( C_\delta(x) \big) = \lim_{n\rightarrow\infty} \mathrm P\big( \pi_n^{-1}(C_\delta(\pi_n(x)) \big) $$

The properties of the premeasure like $\sigma$-additivity are passed through the limit, as eventually inherited by the measures $\mathrm P_n$. The induced outer measure of $\mathrm P_\infty$ can then be restricted to $\mathcal B(C([0,\infty), \mathbb R^d))$, and I believe this will be your desired measure $\mathrm P$. In proving the restriction is a measure, I believe you will need your consistency property.