adapted process

Let $\{X_t\mid t\in T\}$ be a stochastic process defined on a probability space $(\mathrm{\Omega },ℱ,P)$ and $\{{ℱ}_t\mid t\in T\}$ a filtration (an increasing sequence of sigma subalgebras of $ℱ$), where $T$ is a linearly ordered subset of $ℝ$ with a minimum $t_0$. Then the process $\{X_t\}$ is said to be adapted to the filtration $\{{ℱ}_t\}$ if for each $t\ge t_0$, $X_t$ is ${ℱ}_t$-measurable (http://planetmath.org/MathcalFMeasurableFunction):

$$X_t^{-1}(B)\in {ℱ}_t\text{ for each Borel set }B\in ℝ.$$

A stochastic process is an adapted process if it is adapted to some filtration.