Theorem (Left Continuous Adapted = Predictable)

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.

Theorem (Itô Stochastic Integral is a Local Martingale)

Let $M$ be a Local Martingale. Let $X$ be a continuous adapted process (i.e. predictable). Then the Itô Stochastic Integral process $$\left( \int\limits \mathbb{1}_{[0,t]}X \, dM \right)_{t\ge 0}$$is defined and is a continuous local martingale.