Definition 3.1.1 (Previsible $\sigma$-Algebra).label Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the previsible $\sigma$-algebra $\mathscr{J}_{\omega}$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by

In other words, it is the smallest $\sigma$-algebra on $(0, \infty) \times \Omega$ on which every $\bracs{\mathcal{F}_t}$-adapted process with left-continuous sample paths is measurable.