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 in $L^1$ norm, and $\mu$-almost surely.

Proof
By Tower Property of Conditional Expectation:
 * $\expect {f \mid \FF_n} = \expect {\expect {f \mid \FF_\infty} \mid \FF_n}$

Let $\tilde f := \expect {f \mid \FF_\infty}$.

Then we need to show that:
 * $\ds \lim_{n \mathop \to \infty} \expect {\tilde f \mid \FF_n} = \tilde f$

holds in $L^1$ norm, and $\mu$-almost surely.

To this end, let $\epsilon > 0$.

By Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space, there exist:
 * $N \in \N$
 * $g \in \map {L^1} {X, \FF_N, \mu}$

such that:
 * $(1):\quad \norm {\tilde f - g}_1 \le \epsilon$

Then, for all $n \ge N$:

Thus:
 * $\ds \lim_{n \mathop \to \infty} \norm {\expect {\tilde f \mid \FF_n} - \tilde f}_1 = 0$

On the other hand:

Observe:

From $(2)$ and $(3)$:

Therefore: