Increasing Martingale Theorem

Theorem
Let $\struct {X, \Sigma, \mu}$ be a probability space.

Let $f$ be a $\mu$-integrable function.

Given sub-$\sigma$-algebra $\CC \subseteq \Sigma$, let $\expect {f \mid \CC}$ denote the conditional expectation of $f$ on $\CC$.

Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a filtration of $\Sigma$.

Let $\FF_\infty$ be the limit of $\sequence {\FF_n}_{n \mathop \in \N}$.

Then:
 * $\ds \lim_{n \mathop \to \infty} \expect {f \mid \FF_n} = \expect {f \mid \FF_\infty}$

holds $\mu$-almost surely.