Lemma 3.1.4.label Let $(\Omega, \bracs{\cf_t})$ be a filtered space and $X: \Omega \to C([0, \infty); \real^{d})$ be a $\bracs{\mathcal{F}_t}$-adapted process with continuous sample paths, then

1. (1)

   $X$ is $(\mathscr{J}_{\Omega}, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable.

2. (2)

   For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^{d})$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible.