Predictable sigma-algebra ($\mathcal{P}$)

The predictable sigma-algebra on $\Omega \times [0,\infty )$ is generated by $A\times \{0\}$ for $A\in {\mathcal{F}}_0$ and by $A\times (s,t]$ for $A\in {\mathcal{F}}_s$. Equivalently, it is the smallest sigma-algebra making every left-continuous adapted process measurable.

A predictable rectangle has the displayed form for $0\le s<t$. Its indicator is a left-continuous adapted process, and these rectangles generate the predictable sigma-algebra on positive times. If the time-zero slice is included, also use $B\times \{0\}$ with $B\in {\mathcal{F}}_0$.

On $\Omega =(0,\infty )$, observe only $X_t(\omega )=1_{\{t\le \omega \}}$. Before and at the observed lifetime, the natural filtration cannot distinguish any two surviving outcomes. The predictable sigma-algebra on positive times therefore consists exactly of sets whose trace on $D=\{\omega \ge t\}$ is $D\cap (\Omega \times T)$ for a Borel set $T$, and whose trace on $D^c$ is arbitrary product-Borel. To prove sufficiency, $D$ is predictable because $X$ is left-continuous and adapted; on $D^c$, insert a rational $s$ with $\omega <s<t$ and use the predictable rectangles $(E\cap (0,s))\times (s,\infty )$.

For a predictable process $H$ and deterministic $t>0$, $H_t$ is measurable for the left-limit sigma-algebra ${\mathcal{F}}_{t-}$. This follows by taking sections of the generating predictable rectangles. At time zero use ${\mathcal{F}}_{0-}={\mathcal{F}}_0$.

An elementary predictable process may use finitely many ordered stopping times ${\tau }_j$, with bounded ${\mathcal{F}}_{{\tau }_j}$-measurable coefficients $h_j$ on $({\tau }_j,{\tau }_{j+1}]$. Its elementary stochastic integral is ${\sum }_j{h}_j(X_{t\wedge {\tau }_{j+1}}-X_{t\wedge {\tau }_j})$. The stopped indicators are left-continuous and adapted, so the process is predictable; common refinement shows the sum is independent of its representation. Deterministic endpoints recover the usual simple predictable process. These integrands give the precise good-integrator test in the Bichteler-Dellacherie theorem.

A simple predictable process is constant on finitely many deterministic time intervals $(t_k,t_{k+1}]$, and its value there is ${\mathcal{F}}_{t_k}$-measurable. Such processes generate the predictable sigma-algebra and are dense in the corresponding $L^p$ spaces for finite measures.

For every finite measure $\mu$ on the predictable sigma-algebra, bounded simple predictable processes are dense in $L^2(\mu )$. Include a bounded ${\mathcal{F}}_0$-measurable value supported at time zero when $\mu$ may charge that slice. Indicators of the generating rectangles and the Pi-lambda theorem prove density. This gives the completion step in the Itô isometry construction.

Every deterministic càdlàg function is Borel measurable in time. Since the predictable sigma-algebra contains every set $\Omega \times A$ with $A$ Borel, the corresponding deterministic process is predictable.