A Stochastic Process is called progressive or progressively measurable if seen as the mapping Misplaced &H: (\Omega \times [0,T], \mathcal{F}_T \otimes \mathcal{B}([0,T])) &\to (\mathbb{R}, \mathcal{B}) \\ (t, \omega) &\mapsto X_t(\omega) \end{aligned}$$ is measurable with respect to the **progressive $\sigma$-algebra** $$\mathcal{F}_T \otimes \mathcal{B}([0,T]) = \sigma(A \times B \space | \space A \in \mathcal{F}_T, B \in \mathcal{B}([0,T]))$$ $\forall \space T \geq 0$. Note that any (right *or* left) continuous and [[Adapted Process]] is progressive. This can be seen by defining the discretized process: $$X_s^n := \begin{cases} X_{\frac{kt}{n}} & s \in [\frac{(k-1)t}{n}, \frac{kt}{n}) \\ X_t & \text{else} \\ \end{cases}$$ and taking the limit by dominated convergence. Progressive processes are those which are still measurable at an (a.s. finite) [[Stopping Time]]: $$\omega \mapsto X_{T(\omega)}(\omega) \space \text{is} \space \mathcal{F}_T \text{-measurable}$$