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

Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a filtration of $\Sigma$. 
Let $\FF_\infty$ be the limit of $\sequence {\FF_n}_{n \mathop \in \N}$.
Let $p \ge 1$.
Let $\map {L^p} {\cdot}$ denote the $L^p$ spaces.
Then $\ds \bigcup_{n \mathop \ge 0} \map {L^p} {X, \FF_n, \mu}$ is a dense subset of $\map {L^p} {X, \FF_\infty, \mu}$.

Proof:
First, it is a subset, since
:$\forall n \ge 0 : \map {L^p} {X, \FF_n, \mu} \subseteq  \map {L^p} {X, \FF_\infty, \mu}$
in view of $\FF_n \subseteq \FF_\infty$.
In the following, we show its density.
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:
:$\ds \BB := \bigcap_{\epsilon \mathop > 0} \bigcup_{A \mathop \in \AA_0} \set {B \in \Sigma : \map \mu {A \symdif B} \le \epsilon}$
is a $\sigma$-algebra.
In particular:
:$(1):\quad \FF_\infty = \map \sigma {\AA_0} \subseteq \BB$
Let $f \in \map {L^p} {X, \FF_\infty, \mu}$.
Let $\epsilon > 0$.
By Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space, there exist:
:$n \in \Z_{>0}$
:$\alpha_1, \ldots, \alpha_n \in \R_{\ne 0}$
:$B_1, \ldots, B_n \in \FF_\infty$
such that:
:$\ds (2): \quad \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p \le \epsilon$
By $(1)$, there exist:
:$A_1, \ldots, A_n \in \AA_0$
such that:
:$\ds (3): \quad \forall k \in \set {1, \ldots, n} : \map \mu {A_k \symdif B_k} \le \paren {\frac \epsilon {\size {n \alpha _k} } }^p$
By definition of $\AA_0$, for each $k$ there exists an $n_k \ge 0$ such that:
:$A_k \in \FF_{n_k}$
Let $N := \max \set {n_1, \ldots, n_k}$.
Then:
:$A_1, \ldots, A_n \in \FF_N$
In particular:
:$\ds \sum_{k \mathop = 1}^n \alpha_k \chi_{A_k} \in \map {L^p} {X, \FF_N, \mu}$
Moreover:
{{begin-eqn}}
{{eqn | l = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{A_k} }_p
      | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} + \sum_{k \mathop = 1}^n \alpha_k \paren {\chi_{B_k} - \chi_{A_k} } }_p
}}
{{eqn | o = \le
      | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p + \sum_{k \mathop = 1}^n \norm {\alpha_k  \paren {\chi_{B_k} - \chi_{A_k} } }_p
}}
{{eqn | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p + \sum_{k \mathop = 1}^n \size {\alpha_k} \norm { \chi_{B_k} - \chi_{A_k} }_p
}}
{{eqn | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p + \sum_{k \mathop = 1}^n \size {\alpha_k} \paren {\int \size {\chi_{B_k} - \chi_{A_k} }^p \rd \mu }^{\frac 1 p}
      | c = {{Defof|Lp Norm|$L^p$-norm}}
}}
{{eqn | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p + \sum_{k \mathop = 1}^n \size {\alpha_k} \paren {\int \chi_{A_k \symdif B_k} \rd \mu }^{\frac 1 p}
}}
{{eqn | r = \norm {f - \sum_{k \mathop = 1}^n \alpha_k \chi_{B_k} }_p + \sum_{k \mathop = 1}^n \size {\alpha_k} \map \mu {A_k \symdif B_k}^{\frac 1 p}
      | c = {{Defof|Integral of Positive Simple Function}}
}}
{{eqn | o = \le
      | r = \epsilon + \sum_{k \mathop = 1}^n \size {\alpha_k} \frac \epsilon {\size {n \alpha _k} }
      | c = by $(2)$ and $(3)$
}}
{{eqn | r = 2 \epsilon
}}
{{end-eqn}}
{{finish|Details}}
Category:Filtrations of Sigma-Algebras
Category:Lp Spaces
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