Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space

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

Let $\sequence {\FF_n}_{n \ge 0}$ be a discrete-time filtration of $\Sigma$.

Let $\FF_\infty$ be the limit filtration of $\sequence {\FF_n}_{n \ge 0}$.

Let $\map {L^1} {\cdot}$ denote the $L^1$ spaces.

Then $\bigcup_{n \ge 0} \map {L^1} {X, \FF_n, \mu}$ is a dense subset of $\map {L^1} {X, \FF_\infty, \mu}$.

Proof
Let:
 * $\ds \AA_0 := \bigcup_{n \mathop \ge 0} \FF_n$

Then $\AA_0$ is an algebra.

By Sigma-Algebra extended from Algebra by Measure:
 * $\ds \BB := \bigcap_{\epsilon \mathop > 0} \bigcup_{A \in \AA_0} \set { B \in \Sigma : \map \mu {A \symdif B} < \epsilon}$

is a $\sigma$-algebra.

In particular:
 * $\FF_\infty \subseteq \BB$