Progressive measurability

A process $H$ is progressively measurable when, for every $T$, its restriction to $[0,T]\times \Omega$ is measurable for $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$. Continuous adapted processes are progressively measurable, and a Borel function of such a process remains so. A progressively measurable square-integrable Brownian integrand has an equivalent predictable integrand for the Itô integral, up to equality for $dtd\mathbb{P}$.