Wat is Filtration, adapted process?

A filtration is a family $\mathbb F = (\mathcal F_t)_{t \in \mathbb T}$ of sub-$\sigma$-algebras of $\mathcal F$, increasing in $t$: $\mathcal F_s \subseteq \mathcal F_t$ for $s \le t$. The index set $\mathbb T$ is $\{0, 1, \dots, n\}$ or $[0, T]$; in continuous time the filtration is assumed right-continuous and complete (the usual conditions). A process $(X_t)$ is an adapted process if each $X_t$ is $\mathcal F_t$-measurable: its value at $t$ is known at $t$.