Theorem (Lévy’s upward theorem)

Let $Y\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$. Let $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},P)$. Then $$E[Y|\mathcal{F}_{n}]\to E[Y|\mathcal{F}_{\infty^{-}}]$$a.s. and in L1 as $n\to\infty$. Where $\mathcal{F}_{\infty^{-}}=\bigcup_{n\in\mathbb{N}}\mathcal{F}_{n}$.

Theorem (Lévy’s downward theorem)

Let $Y\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$. Let $(\mathcal{F}_{n})_{n\in\mathbb{Z}^{-}}$ be a filtration on $(\Omega,\mathcal{F},P)$. Then $$E[Y|\mathcal{F}_{n}]\to E[Y|\mathcal{F}_{-\infty}]$$a.s. and in L1 as $n\to-\infty$. Where $\mathcal{F}_{-\infty}=\bigcap_{n\in\mathbb{Z}^{-}}\mathcal{F_{n}}$.