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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.^{[1]} Progressively measurable processes are important in the theory of Itô integrals.

Let

- be a probability space;
- be a measurable space, the
*state space*; - be a filtration of the sigma algebra ;
- be a stochastic process (the index set could be or instead of );
- be the Borel sigma algebra on .

The process is said to be **progressively measurable**^{[2]} (or simply **progressive**) if, for every time , the map defined by is -measurable. This implies that is -adapted.^{[1]}

A subset is said to be **progressively measurable** if the process is progressively measurable in the sense defined above, where is the indicator function of . The set of all such subsets form a sigma algebra on , denoted by , and a process is progressively measurable in the sense of the previous paragraph if, and only if, it is -measurable.

- It can be shown
^{[1]}that , the space of stochastic processes for which the Itô integral

- with respect to Brownian motion is defined, is the set of equivalence classes of -measurable processes in .

- Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.
^{[1]} - Every measurable and adapted process has a progressively measurable modification.
^{[1]}

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^{a}^{b}^{c}^{d}^{e}Karatzas, Ioannis; Shreve, Steven (1991).*Brownian Motion and Stochastic Calculus*(2nd ed.). Springer. pp. 4–5. ISBN 0-387-97655-8. **^**Pascucci, Andrea (2011). "Continuous-time stochastic processes".*PDE and Martingale Methods in Option Pricing*. Bocconi & Springer Series. Springer. p. 110. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1. S2CID 118113178.