Progressively measurable process

In mathematics, progressive measurability is a property in the theory of . A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itō integrals.

Let

The process ${\displaystyle X}$ is said to be progressively measurable (or simply progressive) if, for every time ${\displaystyle t}$, the map ${\displaystyle [0,t]\times \Omega \to \mathbb {X} }$ defined by ${\displaystyle (s,\omega )\mapsto X_{s}(\omega )}$ is ${\displaystyle \mathrm {Borel} ([0,t])\otimes {\mathcal {F}}_{t}}$-measurable. This implies that ${\displaystyle X}$ is ${\displaystyle {\mathcal {F}}_{t}}$-adapted.

...
Wikipedia