Density of elementary predictable processes in predictable L2

Put $\mathcal{D}:=\{A\in {\mathcal{P}}_T:1_A\in \bar{\mathcal{E}}\}$, where the closure is taken in the $L^2(\mu )$ metric of [F2]. By [F1] the set $\bar{\mathcal{E}}$ contains every constant $L^2(\mu )$ class (represented elementarily by its value on $(0,T]$ and by $0$ at time $0$) and is closed under finite linear combinations of its elements: if $U_1,\dots ,U_k\in \bar{\mathcal{E}}$ and $a_1,\dots ,a_k\in \mathbb{R}$, choose $G_i\in \mathcal{E}$ with $\mathrm{\parallel }{U}_i-G_i{\mathrm{\parallel }}_2<\delta$ and use the triangle inequality to bound $\mathrm{\parallel }{\sum }_i{a}_i{U}_i-{\sum }_i{a}_i{G}_i{\mathrm{\parallel }}_2$ by $\delta {\sum }_i\mathrm{\mid }{a}_i\mathrm{\mid }$.

Every generator of ${\mathcal{P}}_T$ lies in $\mathcal{D}$, and $\mathcal{D}$ contains $\mathrm{\varnothing }$ and $[0,T]\times \mathrm{\Omega }$. Indeed, for $(s,u]\times C$ with $s>0$ the process $1_C{1}_{(s,u]}$ is elementary on the partition $0<s<u<T$ when $u<T$, with coefficients $0,1_C,0$, and on the partition $0<s<u=T$ when $u=T$, with coefficients $0,1_C$; all coefficients are measurable at the left endpoints since $1_C\in {\mathcal{F}}_s$ and constants are ${\mathcal{F}}_0$-measurable. For a generator $(0,u]\times C$ with $C\in {\mathcal{F}}_0$, use coefficients $1_C,0$ on $0<u<T$ when $u<T$, and the single coefficient $1_C$ when $u=T$. For $\{0\}\times C$ the zero process is elementary with $\mathrm{\parallel }{1}_{\{0\}\times C}-0{\mathrm{\parallel }}_2^2=\mu (\{0\}\times C)=0$ by [F7]. The one-block elementary process $1_{(0,T]}$ represents $1_{[0,T]\times \mathrm{\Omega }}$ in $L^2(\mu )$ because the omitted time-zero slice is null, so the whole space belongs to $\mathcal{D}$; the zero elementary process represents $1_{\mathrm{\varnothing }}$, so $\mathrm{\varnothing }\in \mathcal{D}$.

$\mathcal{D}$ is closed under complements: if $1_A$ is approximated by bounded elementary $G^n$, then the bounded elementary processes $1_{(0,T]}-G^n$ represent the classes $1-G^n$ and converge to $1-1_A=1_{A^c}$ in $L^2(\mu )$ by the linearity of [F1] and the norm axioms of [F2].

$\mathcal{D}$ is closed under countable unions of pairwise disjoint sets: let $A_1,A_2,\cdots \in \mathcal{D}$ be pairwise disjoint with union $A$. Given $\epsilon >0$, choose $m$ so large that $({\sum }_{j>m}\mu (A_j){)}^{1\mathrm{/}2}<\epsilon \mathrm{/}2,$ which is possible because the disjoint additivity of the finite measure $\mu$ gives ${\sum }_j\mu (A_j)=\mu (A)\le T<\mathrm{\infty }$; since each $A_j$ lies in $\mathcal{D}$, choose for $j\le m$ a bounded elementary $G_j$ with $\mathrm{\parallel }{1}_{A_j}-G_j{\mathrm{\parallel }}_2<\epsilon \mathrm{/}(2m)$; the finite sum $S:={\sum }_{j\le m}{G}_j$ is bounded and elementary on a common refinement by [F1]; the complement of ${\bigcup }_{j\le m}{A}_j$ inside $A$ is ${\bigcup }_{j>m}{A}_j$, so [F2] and the triangle inequality give $\mathrm{\parallel }{1}_A-S{\mathrm{\parallel }}_2\le (\mu ({\bigcup }_{j>m}{A}_j){)}^{1\mathrm{/}2}+{\sum }_{j=1}^m\mathrm{\parallel }{1}_{A_j}-G_j{\mathrm{\parallel }}_2<\epsilon \mathrm{/}2+\epsilon \mathrm{/}2=\epsilon$, hence $A\in \mathcal{D}$.

The family of finite intersections of generators of ${\mathcal{P}}_T$ is the pi-system generated by the generators, and it is contained in $\mathcal{D}$: by [F3] a finite intersection is empty, a generator, or a time-zero generator, and $\mathcal{D}$ contains $\mathrm{\varnothing }$ and every generator by step 1.2.

The family $\mathcal{D}$ contains $\mathrm{\varnothing }$, the whole space $[0,T]\times \mathrm{\Omega }$ and is closed under complements (step 2.1) and under countable disjoint unions (step 2.2), so it is a lambda-system; step 2.3 shows that it contains the pi-system of finite intersections of generators, whose generated sigma-algebra is ${\mathcal{P}}_T$ by [F3].

By Dynkin's pi-lambda theorem [F4], ${\mathcal{P}}_T=\sigma (\text{generators})\subseteq \mathcal{D}$. Hence the indicator of every predictable set is an $L^2(\mu )$ limit of bounded elementary processes.

Consequently every bounded ${\mathcal{P}}_T$-measurable simple function ${\sum }_{i=1}^k{a}_i{1}_{A_i}$ is in $\bar{\mathcal{E}}$: approximate each indicator $1_{A_i}$ by bounded elementary processes and combine the finitely many approximants with the coefficients $a_i$ using the closure under finite linear combinations from step 1.1.

Let $H\ge 0$ be ${\mathcal{P}}_T$-measurable with $H\in L^2(\mu )$. By [F5] choose nonnegative simple ${\mathcal{P}}_T$-measurable $s_n\uparrow H$ pointwise; each $s_n$ is bounded and lies in $\bar{\mathcal{E}}$ by step 5.1. Since $0\le s_n\le H$ and $H\in L^2(\mu )$, [F6] gives $\mathrm{\parallel }H-s_n{\mathrm{\parallel }}_2\to 0$, and for $\epsilon >0$ one takes $n$ with $\mathrm{\parallel }H-s_n{\mathrm{\parallel }}_2<\epsilon \mathrm{/}2$ and then a bounded elementary $G$ with $\mathrm{\parallel }{s}_n-G{\mathrm{\parallel }}_2<\epsilon \mathrm{/}2$, so that $\mathrm{\parallel }H-G{\mathrm{\parallel }}_2<\epsilon$ by the triangle inequality.

For a general real $H\in L^2(\mu )$, decompose $H=H^{+}-H^{-}$; both parts are nonnegative and ${\mathcal{P}}_T$-measurable with $H^{\pm }\le \mathrm{\mid }H\mathrm{\mid }$, so both lie in $L^2(\mu )$ and each is approximated to within $\epsilon \mathrm{/}2$ by a bounded elementary process by step 6.1; their difference is bounded and elementary on a common refinement by [F1] and is within $\epsilon$ of $H$ by the triangle inequality.

Steps 6.1 and 7.1 prove the density statement for nonnegative and for general predictable $L^2$ classes, with all approximants bounded and elementary; equalities of processes are equalities of $(\mathrm{d}t\otimes P)$-classes throughout, which is exactly the qualification "modulo product-almost-everywhere equality". For each fixed $m$, step 2.2 selects only the finite list $G_1,\dots ,G_m$, so the density argument itself uses no choice principle; the declared AC is inherited from the standing hypothesis (H) and ambient probability interfaces as recorded in [F8].