Relative sizes of Skorokhod and product topologies on space of sample paths

Question

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg functions (i.e. continuous from the right, limits exist from the left).

On $C$ we can define the Skorokhod metric; and this metric induces the corresponding Borel sigma algebra.

On the other hand, we have the usual product sigma algebra on the space $E^T$, and restricting this sigma algebra to $C$ in the natural way yields a second sigma algebra on the space $C$.

My question is this: Is there any relation in size (with respect to inclusion) of these two sigma algebras defined on $C$.

Many thanks for your help.

Answer 1

Proposition 7.1 in Chapter 3 of Markov Processes: Characterization and Convergence by Stewart N. Ethier and Thomas G. Kurtz says the following: $${\cal B}(D_E[0,\infty)) \supset \sigma(\pi_t: 0\leq t<\infty) $$ with equality when the metric space $E$ is separable.

Answer 2

First of all, I have to mention that the following results hold in the case $E \subseteq \mathbb{R}^d$, but I'm unfortunately not aware whether they can be generalized to an arbitrary compact metric space $E$.

Billingsley [1, Theorem 16.6] proves that

$$\sigma(\pi_t|_{D[0,\infty)}; t \geq 0) = \mathcal{B}(D[0,\infty))$$

where $\mathcal{B}(D[0,\infty))$ denotes the $\sigma$-algebra generated by the Skorohod metric on $D[0,\infty)$ and $\pi_t$ the canonical projection. This implies in particular that their restriction to $C[0,\infty)$ coincides. Moreover, one can show that this $\sigma$-algebra equals the $\sigma$-algebra generated by the local uniform metric

$$d(f,g) := \sum_{n=1}^{\infty} 2^{-n} \cdot \left( 1 \wedge \sup_{0 \leq t \leq n} |f(t)-g(t)| \right)$$

see e.g. [1] or [2, Lemma 4.1].

[1] Patrick Billingsley: Convergence of probability measures.

[2] René L. Schilling/Lothar Partzsch: Brownian motion - An Introduction to Stochastic Processes.