Let $\{X_{t}\mid t\in T\}$ be a stochastic process defined on a probability space $(\Omega,\mathcal{F},P)$ and $\{\mathcal{F}_{t}\mid t\in T\}$ a filtration (an increasing sequence of sigma subalgebras of $\mathcal{F}$), where $T$ is a linearly ordered subset of $\mathbb{R}$ with a minimum $t_{0}$. Then the process $\{X_{t}\}$ is said to be adapted to the filtration $\{\mathcal{F}_{t}\}$ if for each $t\geq t_{0}$, $X_{t}$ is $\mathcal{F}_{t}$-measurable (http://planetmath.org/MathcalFMeasurableFunction):
 $X_{t}^{-1}(B)\in\mathcal{F}_{t}\mbox{ for each Borel set }B\in\mathbb{R}.$