Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ is measurable with $\mathcal F_i$ (adaptable). Is random variable $X_i$ also measurable on $\mathcal {F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Since $\sigma(X_i)\subset\mathcal F_i\subset\mathcal F_{i+1}$, $X_i$ is certainly $\mathcal F_{i+1}$-measurable. 

But in general $X_i$ is not necessarily $\mathcal F_{i-1}$ measurable: take $\mathcal F_i:=\sigma(X_k,1\leqslant k\leqslant i)$ where the $X_k$ are independent integrable non degenerated random variables. Then $\mathbb E(X_i\mid\mathcal F_{i-1})=\mathbb E(X_i)\neq X_i$. 

It justifies the notion of predictable sequence/process.
If $\mathcal{F}_i \subset \mathcal{F}_{i+1} \subset \mathcal{F}_{i+2}$ for $i = 1, 2, \ldots$, $X_i$ is $\mathcal{F}_{i+1}$-measurable, but not $\mathcal{F}_{i-1}$-measurable.

[ incorrect statements removed. ]