# Showing that the elementary processes generate the predictable sigma-algebra

We work with respect to a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},P)$

An elementary process is of the form

\begin{equation} \label{eq:1} \xi(t) = Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_k<t\le t_k\}} \end{equation} for ${n\ge 0}$, times ${s_k<t_k}$, ${\mathcal{F}_0}$-measurable random variable ${Z_0}$ and ${\mathcal{F}_{s_k}}$-measurable random variables ${Z_k}$.

The predictable sigma-algebra on ${{\mathbb R}^+\times\Omega}$ is the sigma-algebra generated by the left-continuous and adapted processes.

How can we show that the elementary processes also generate the predictable sigma-algebra ?

• This is pretty standard exercise. Where do you get stuck? The other inclusion is trivial at least. Feb 18, 2018 at 21:15
• I really have no idea how to show it. My analysis background tends to tell me that we can approximate left-continuous and adapted processes by elementary processes, that the elementary processes are dense in the left-continuous and adapted processes in some sense. But we would need a topology on the left-continuous and adapted processes. What topology one should use ? Ucp topology ? Feb 18, 2018 at 21:46
• You need to show that every left continuous and adapted process is an elementary process and that elementary processes are left continuous and adapted. Then you are done. Feb 18, 2018 at 22:32
• Can you give a hint for the non-trivial part ? Feb 18, 2018 at 23:16
• Sorry for the long response. Feb 21, 2018 at 11:40

Denote the sigma-algebra generated by predictable processes by $\mathcal{E}$ and the sigma-algebra generated by left-continuous and adapted processes by $\mathcal{H}$. We want to show that $\mathcal{E} = \mathcal{H}$.
Clearly the elementary processes $$\xi(t) = Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_k<t\le t_k\}}$$ are left-continuous and adapted (and also constant on the intervals $(t_k,t_{k+1})$). Thus at leas we have $\mathcal{E} \subset \mathcal{H}$ i.e. every predictable process is left-continuous and adapted.
Conversely if $X$ is adapted and left-continuous process it is a limit of piecewise constant functions. More presicely $X$ is the limit (as $n \to \infty$) of $$X_t = X_01_{\{t=0\}}+\sum_{i=1}^\infty X_{\{(i-1)/n\}} 1_{\{(i-1)/n < t \leq i/n\}}.$$ Thus also $\mathcal{H} \subset \mathcal{E}$.