= Survival-observation predictable sigma-algebra
{title2=$\mathcal P$}
On $\Omega=(0,\infty)$, observe only $X_t(\omega)=\mathbf1_{\{t\leq\omega\}}$. 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 $D=\{\omega\geq t\}$ 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 $(E\cap(0,s))\times(s,\infty)$.
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