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In discrete time, a process $A_n$ is predictable or previsible when $A_n$ is ${\mathcal{F}}_{n-1}$-measurable for every $n\ge 1$. In continuous time, predictability means measurability with respect to the predictable sigma-algebra.

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 $B\in {\mathcal{F}}_0$.

For a L2-bounded continuous martingale $M$, define a finite measure on the predictable sigma-algebra by ${\nu }_M(A)=\mathbb{E}{\int }_0^{\infty }{1}_{A}d[M]$. Its total mass is $\mathbb{E}[M{]}_{\infty }=\mathbb{E}{M}_{\infty }^2-\mathbb{E}{M}_0^2$. The Lebesgue space $L^2({\nu }_M)$ identifies predictable processes that agree ${\nu }_M$-almost everywhere. Integration against $M$ is an isometry from this space into L2-bounded continuous martingales starting at zero by the Itô isometry. This gives the correct integrand space even when the quadratic variation is random or is not absolutely continuous in time.

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 $(E\cap (0,s))\times (s,\infty )$.