Progressively measurable process
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 Itō integrals.
Definition
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 ).
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.
Properties
 It can be shown^{[1]} that , the space of stochastic processes for which the Ito integral

 with respect to Brownian motion is defined, is the set of equivalence classes of measurable processes in .
 Every adapted process with left or rightcontinuous 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]}
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. pp. 4–5. ISBN 0387976558.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Pascucci, Andrea (2011) PDE and Martingale Methods in Option Pricing. Berlin: Springer^{[page needed]}
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