Lévy's upward theorem

Remarks

The sequence $M_n = E[X \mid \mathcal{F}_n]$ is the Doob martingale of $X$, and it is uniformly integrable. By Doob's martingale convergence theorems it converges almost surely and in $L^1$ to some $M_\infty$, which is $\mathcal{F}_\infty$-measurable. For $A \in \mathcal{F}_m$ and $n \ge m$, $E[M_n \mathbf{1}_A] = E[X \mathbf{1}_A]$, and the $L^1$ limit gives $E[M_\infty \mathbf{1}_A] = E[X \mathbf{1}_A]$. A π–λ argument extends this to all $A \in \mathcal{F}_\infty$, so $M_\infty = E[X \mid \mathcal{F}_\infty]$. Taking $X = \mathbf{1}_A$ with $A \in \mathcal{F}_\infty$ gives Lévy's zero–one law: $P(A \mid \mathcal{F}_n) \to \mathbf{1}_A$ almost surely.

Going deeper

For a student

The theorem is a probabilistic counterpart of the Lebesgue differentiation theorem. On $[0, 1)$ with the dyadic filtration, $E[X \mid \mathcal{F}_n]$ is the average of $X$ over the dyadic interval of length $2^{-n}$ containing the point. Lévy's zero–one law follows at once, and it gives Kolmogorov's zero–one law for tail events of independent sequences. A companion result, Lévy's downward theorem, treats decreasing families of σ-algebras.