Theorem: Increasing Martingale 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 $n \ge N$:
{{begin-eqn}}
{{eqn | l = \norm {\expect {\tilde f \mid \FF_n} - \tilde f}_1
      | r = \norm {\expect {\tilde f - g \mid \FF_n} - \paren {\tilde f - g} }_1
}}
{{eqn | o = \le
      | r = \norm {\expect {\tilde f - g \mid \FF_n} }_1 + \norm {\tilde f - g}_1
}}
{{eqn | o = \le
      | r = 2 \norm {\tilde f - g}_1
}}
{{eqn | o = \le
      | r = 2 \epsilon
}}
{{end-eqn}}
Thus:
:$\ds \lim_{n \mathop \to \infty} \norm {\expect {\tilde f \mid \FF_n} - \tilde f}_1 = 0$
{{qed|lemma}}
On the other hand:
{{begin-eqn}}
{{eqn | l = \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}
      | r = \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f - g \mid \FF_n} - \paren {\tilde f - g} } > \sqrt \epsilon}
}}
{{eqn | o = \le
      | r = \map \mu {\set {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } \cup \set { \size { \tilde f - g} > \frac {\sqrt \epsilon} 2 } }
}}
{{eqn | o = \le
      | r = \map \mu {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } + \map \mu { \size { \tilde f - g} > \frac {\sqrt \epsilon} 2 }
}}
{{eqn | o = \le
      | r = \map \mu {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } + \frac 2 {\sqrt \epsilon} \norm {\tilde f - g}_1
      | c = Markov's Inequality
      | n = 2
}}
{{end-eqn}}
Observe:
{{begin-eqn}}
{{eqn | l = \map \mu { \sup_{n \in \N} \size{ \expect { {\tilde f - g} \mid \FF_n} } \ge \frac {\sqrt \epsilon} 2}
      | o = \le
      | r = \map \mu { \sup_{n \in \N} \expect {\size {\tilde f - g} \mid \FF_n} \ge \frac {\sqrt \epsilon} 2}
}}
{{eqn | r = \lim_{m \to \infty} \map \mu { \sup_{0 \le n \le m} \expect {\size {\tilde f - g} \mid \FF_n} \ge \frac {\sqrt \epsilon} 2}
      | c = Measure of Limit of Increasing Sequence of Measurable Sets
}}
{{eqn | o = \le
      | r = \lim_{m \to \infty} \frac 2 {\sqrt \epsilon} \expect {\expect {\size {\tilde f - g} \mid \FF_m} }
      | c = Doob's Maximal Inequality
}}
{{eqn | r = \frac 2 {\sqrt \epsilon} \expect {\size {\tilde f - g} }
      | c = {{Defof|Conditional Expectation on Sigma-Algebra}}
}}
{{eqn | r = \frac 2 {\sqrt \epsilon} \norm {\tilde f - g}_1
      | c = {{Defof|Lp Norm|$L^1$ norm}}
      | n = 3
}}
{{end-eqn}}
From $(2)$ and $(3)$:
{{begin-eqn}}
{{eqn | l = \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}
      | o = \le
      | r = \frac 4 {\sqrt \epsilon} \norm {\tilde f - g}_1
}}
{{eqn | o = \le
      | r = 4 \sqrt \epsilon
      | c = by $(1)$
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > 0}
      | r = \lim_{\epsilon \mathop \to 0} \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}
      | c = Measure of Limit of Increasing Sequence of Measurable Sets
}}
{{eqn | o = \le
      | r = \lim_{\epsilon \mathop \to 0} 4 \sqrt \epsilon
}}
{{eqn | r = 0
}}
{{end-eqn}}
{{finish|details}}