Let
- $(\Omega,\mathcal A)$ be a measurable space;
- $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $$\mathcal F_\infty:=\sigma(\mathcal F_t,t\ge0);$$
- $\tau:\Omega\to[0,\infty]$ and $$\mathcal F_{\tau-}:=\sigma(\mathcal F_0\cup\{F\cap\{t<\tau\}:F\in\mathcal F_t\text{ and }t>0\});$$
- $$\overline{\mathcal F}^{pp}:=\underbrace{\bigcup_{F\in\mathcal F_0}F\times\{0\}\cup\bigcup_{0\le s<t}\bigcup_{F\in\mathcal F_s}F\times(s,t]}_{=:\:\mathcal F^{pp}}\cup\bigcup_{F\in\mathcal F_\infty}F\times\{\infty\}$$ and \begin{align}\mathcal F^p&:=\sigma(\mathcal F^{pp})\\\overline{\mathcal F}^p&:=\sigma(\mathcal F^{pp});\end{align}
- $(E,\mathcal E)$ be a measurable space;
- $(X_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued $(\mathcal F_t)_{t\ge0}$-predictable process on $(\Omega,\mathcal A)$; i.e. $X:\Omega\times[0,\infty)\to E$ is $(\mathcal F^p,\mathcal E)$-measurable;
- $X_\infty:\Omega\to E$ be $\mathcal F_\infty$-measurable.
We easily see that $(X_t)_{t\in[0,\:\infty]}$ is $(\mathcal F_t)_{t\in[0,\:\infty]}$-predictable; i.e. $X:\Omega\times[0,\infty]\to E$ is $(\overline{\mathcal F}^p,\mathcal E)$-measurable.
How can we show that $X_\infty$ is $\mathcal F_{\tau-}$-measurable (maybe under the assumption that $\tau$ is an $(\mathcal F_t)_{t\ge0}$-stopping time)?
Let $$\mathcal H:=\{H:\Omega\times[0,\infty]\to\mathbb R\mid H_\tau\text{ is }\mathcal F_{\tau-}\text{-measurable}\}.$$
My idea is to show that $1_R\in\mathcal H$ for all $R\in\overline{\mathcal F}^p$ and hence $\mathcal H$ contains every $\overline{\mathcal F}^p$-measurable function by the monotone class theorem. It's easy to verify this for $R=\{0\}\times F$ for some $F\in\mathcal F_0$ and for $R=\{\infty\}\times F$ for some $F\in\mathcal F_\infty$, but how do we show it for $R=(s,t]\times F$ for some $F\in\mathcal F_s$ and $0\le s<t$?
Moreover, this only shows the desired result for $(E,\mathcal E)=(\mathbb R,\mathcal B(\mathbb R))$. The monotone class theorem is also applicable if $(\mathbb R,\mathcal B(\mathbb R))$ is replaced by a normed $\mathbb R$-vector space, but what can we do in the case of a general measurable space $(E,\mathcal E)$?