Let

• ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$  be a probability space;
• ${\displaystyle (\mathbb {X} ,{\mathcal {A}})}$  be a measurable space, the state space;
• ${\displaystyle \{{\mathcal {F}}_{t}\mid t\geq 0\}}$  be a filtration of the sigma algebra ${\displaystyle {\mathcal {F}}}$ ;
• ${\displaystyle X:[0,\infty )\times \Omega \to \mathbb {X} }$  be a stochastic process (the index set could be ${\displaystyle [0,T]}$  or ${\displaystyle \mathbb {N} _{0}}$  instead of ${\displaystyle [0,\infty )}$ );
• ${\displaystyle \mathrm {Borel} ([0,t])}$  be the Borel sigma algebra on ${\displaystyle [0,t]}$ .

The process ${\displaystyle X}$  is said to be progressively measurable^[2] (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.^[1]

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.

• It can be shown^[1] that ${\displaystyle L^{2}(B)}$ , the space of stochastic processes ${\displaystyle X:[0,T]\times \Omega \to \mathbb {R} ^{n}}$  for which the Itô integral

${\displaystyle \int _{0}^{T}X_{t}\,\mathrm {d} B_{t}}$ 

with respect to Brownian motion ${\displaystyle B}$  is defined, is the set of equivalence classes of ${\displaystyle \mathrm {Prog} }$ -measurable processes in ${\displaystyle L^{2}([0,T]\times \Omega ;\mathbb {R} ^{n})}$ .

• 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]