This is what I thought initially, just a traditional enlargement of the space and filtration so as to to have a Brownian motion to work with. What is making me think this notion of “adjunction” is different is the context in which the term appears in the theorem. That is, as in the theorem, we only need to adjunct a Brownian motion if the measure (P x Lebesgue) of {(omega, t) : psi_t (omega) = 0} is zero where (psi_t) is the nondecreasing process that defines the 2nd moment of the increments of (X_t) — it’s not clear to me that this has any connection to the underlying filtered probability space not being able to support a Brownian motion, but I don’t know for certain. I will check out Doob’s text, thanks!