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Definition D4109
Predictable random sequence

Let $P = (\Omega, \mathcal{F}, \mathbb{P}, \{ \mathcal{G}_n \}_{n \in \mathbb{N}})$ be a D1726: Filtered probability space.
A D1723: Random sequence $X_0, X_1, X_2, \dots$ on $P$ is a predictable random sequence on $P$ if and only if $$\forall \, n \in \mathbb{N} : \sigma_{\text{pullback}} \langle X_{n + 1} \rangle \subseteq \mathcal{G}_n$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P}, \{ \mathcal{G}_n \}_{n \in \mathbb{N}})$ be a D1726: Filtered probability space.
A D1723: Random sequence $X_0, X_1, X_2, \dots$ on $P$ is a predictable random sequence on $P$ if and only if $$\forall \, n \in \mathbb{N} : X_{n + 1} \in \mathcal{G}_n$$