Let

• ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$  be a probability space;
• ${\displaystyle I}$  be an index set with a total order ${\displaystyle \leq }$  (often, ${\displaystyle I}$  is ${\displaystyle \mathbb {N} }$ , ${\displaystyle \mathbb {N} _{0}}$ , ${\displaystyle [0,T]}$  or ${\displaystyle [0,+\infty )}$ );
• ${\displaystyle \mathbb {F} =\left({\mathcal {F}}_{i}\right)_{i\in I}}$  be a filtration of the sigma algebra ${\displaystyle {\mathcal {F}}}$ ;
• ${\displaystyle (S,\Sigma )}$  be a measurable space, the state space;
• ${\displaystyle X_{i}:I\times \Omega \to S}$  be a stochastic process.

The stochastic process ${\displaystyle (X_{i})_{i\in I}}$  is said to be adapted to the filtration ${\displaystyle \left({\mathcal {F}}_{i}\right)_{i\in I}}$  if the random variable ${\displaystyle X_{i}:\Omega \to S}$  is a ${\displaystyle ({\mathcal {F}}_{i},\Sigma )}$ -measurable function for each ${\displaystyle i\in I}$ .^[2]

Consider a stochastic process X : [0, T] × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets.

• If we take the natural filtration F_•^X, where F_t^X is the σ-algebra generated by the pre-images X_s⁻¹(B) for Borel subsets B of R and times 0 ≤ s ≤ t, then X is automatically F_•^X-adapted. Intuitively, the natural filtration F_•^X contains "total information" about the behaviour of X up to time t.
• This offers a simple example of a non-adapted process X : [0, 2] × Ω → R: set F_t to be the trivial σ-algebra {∅, Ω} for times 0 ≤ t < 1, and F_t = F_t^X for times 1 ≤ t ≤ 2. Since the only way that a function can be measurable with respect to the trivial σ-algebra is to be constant, any process X that is non-constant on [0, 1] will fail to be F_•-adapted. The non-constant nature of such a process "uses information" from the more refined "future" σ-algebras F_t, 1 ≤ t ≤ 2.

• Predictable process
• Progressively measurable process