In mathematics, progressive measurability is a property in the theory of stochastic processes. 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 Itô 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.

A subset ${\displaystyle P\subseteq [0,\infty )\times \Omega }$ is said to be progressively measurable if the process ${\displaystyle X_{s}(\omega ):=\chi _{P}(s,\omega )}$ is progressively measurable in the sense defined above, where ${\displaystyle \chi _{P}}$ is the indicator function of ${\displaystyle P}$. The set of all such subsets ${\displaystyle P}$ form a sigma algebra on ${\displaystyle [0,\infty )\times \Omega }$, denoted by ${\displaystyle \mathrm {Prog} }$, and a process ${\displaystyle X}$ is progressively measurable in the sense of the previous paragraph if, and only if, it is ${\displaystyle \mathrm {Prog} }$-measurable.