Theorem

Let $X:\mathbb{R}^{+}\times\Omega\to \mathbb{R}$. Assume $(X_{t})_{t\ge 0}$ is $(\mathcal{F}_{t})_{t\ge 0}$-adapted and left continuous. Then $X:(\mathbb{R}^{+}\times\Omega,\mathscr{P})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable i.e. $X$ is a predictable process.