A series of random variables $\{X_i\}_{i\in \mathbb{Z}}$, $X_i: \Omega\to \mathbb{R}$ is called $0$-independent if for every $t\in\mathbb{Z}$,
$(\cdots,X_{t-1},X_t)$ and $(X_{t+1},\cdots)$ are independent.
The same series is called independent if for arbitrary subset $I$ of $\mathbb{Z}$,
$(X_i)_{i\in I}$ is independent.
independent $\implies$ $0$-independent is straightforward.
How about the other direction? Should I use marginal distirbution $\mathbb{P}(X_t\in B_t, X_{t+1}\in \Omega,\cdots) = \mathbb{P}(X_t\in B_t)$?