Let $P = (\Omega, \mathcal{F}, \mathbb{P}, \{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space such that

(i)	$\{ X_j \}_{j \in J}$ is a D1721: Random collection on $P$

Then $\{ X_j \}_{j \in J}$ is an adapted random collection on $P$ if and only if \begin{equation} \forall \, j \in J : X_j \in \mathcal{G}_j \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P}, \{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space such that

(i)	$\{ X_j \}_{j \in J}$ is a D1721: Random collection on $P$

Then $\{ X_j \}_{j \in J}$ is an adapted random collection on $P$ if and only if \begin{equation} \forall \, j \in J : \sigma_{\text{pullback}} \langle X_j \rangle \subseteq \mathcal{G}_j \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$\{ \mathcal{G}_j \}_{j \in J}$ is a D3346: Sigma-algebra filtration for $\mathcal{F}$ on $P$
(ii)	$\{ X_j \}_{j \in J}$ is a D1721: Random collection on $P$

Then $\{ (X_j, \mathcal{G}_j) \}_{j \in J}$ is an adapted random collection on $P$ if and only if \begin{equation} \forall \, j \in J : \sigma_{\text{pullback}} \langle X_j \rangle \subseteq \mathcal{G}_j \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$\{ \mathcal{G}_j \}_{j \in J}$ is a D3346: Sigma-algebra filtration for $\mathcal{F}$ on $P$
(ii)	$\{ X_j \}_{j \in J}$ is a D1721: Random collection on $P$

Then $\{ (X_j, \mathcal{G}_j) \}_{j \in J}$ is an adapted random collection on $P$ if and only if \begin{equation} \forall \, j \in J : X_j \in \mathcal{G}_j \end{equation}