A stochastic process $(X_t)_{t\in\mathbb{Z}_+}$ is said to be adapted to a filtration $(\mathcal{F}_t)$ on the measurable space $(\Omega,\mathcal{F})$ if $X_t$ is an $\mathcal{F}_t$ -measurable random variable for each $t=0,1,\ldots$ . However, for continuous-time processes, where the time $t$ ranges over an arbitrary index set $\mathbb{T}\subseteq\mathbb{R}$ , the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map \begin{equation*} X\colon\mathbb{T}\times\Omega\rightarrow\mathbb{R},\ (t,\omega)\mapsto X_t(\omega) \end{equation*}it is useful to consider the measurability of $X$ .

The process $X$ is progressive or progressively measurable if, for every $s\in\mathbb{T}$ , the stopped process $X^s_t\equiv X_{\min(s,t)}$ is $\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_s$ -measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when $\mathbb{T}$ is countable, the converse holds and every adapted process is progressive.

A set $S\subseteq\mathbb{T}\times\Omega$ is said to be progressive if its characteristic function $1_S$ is progressive. Equivalently, \begin{equation*} S\cap\left( (-\infty,s]\times\Omega\right)\in\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_s \end{equation*}for every $s\in\mathbb{T}$ . The progressively measurable sets form a $\sigma$ -algebra, and a stochastic process is progressive if and only if it is measurable with respect to this $\sigma$ -algebra.