Definition (Adapted — discrete)

Let $(X_{n})_{n\in\mathbb{N}}$ be a Stochastic Process on $(\Omega,\mathcal{F},P)$ and $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ a Filtration on $(\Omega,\mathcal{F},P)$. $(X_{n})_{n\in\mathbb{N}}$ is called $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-adapted if $$X_{n}\text{ is }\mathcal{F}_{n}\text{-measurable }\forall n\in\mathbb{N}$$

Definition (Adapted — continuous)

Let $(X_{t})_{t\in\mathbb{R}^{+}}$ be a process on $(\Omega,\mathcal{F},P)$ and let $(\mathcal{F}_{t})_{t\in\mathbb{R}^{+}}$ be a filtration on $(\Omega,\mathcal{F},P)$. $(X_{t})_{t\in\mathbb{R}^{+}}$ is said to be $(\mathcal{F}_t)_{t\ge 0}$ is said to be $(\mathcal{F}_{t})_{t\ge 0}$-adapted if $$X_{t}\text{ is }\mathcal{F}_{t}\text{-measurable }\forall t\in\mathbb{R}^{+}$$