Let $X\_s(\omega)$ be measurable and adapted.


Under what conditions will the process
$$
F\_{t}(\omega) = \int\_0^t X\_s(\omega) \, ds
$$ also be adapted?


To me it seems that adaptedness and measurability should be enough but at the bottom of page 133 in Karatzas and Shreve they say this is not enough. Why?



Is the integral of an adapted, measurable process adapted?
Assuming that $\int\_0^t|X\_s(\omega)|\,ds<\infty$ for all $t>0$ and all $\omega$, and that the filtration satisfies the usual conditions, the process $F\_t:=\int\_0^t X\_s\,ds$ is well defined and adapted (even predictable, being continuous). This matter is discussed in the paper "Un exemple de processus mesurable adapté non progressif" by G. Letta: <https://link.springer.com/chapter/10.1007%2FBFb0084150>. Letta provides an example in which $F$ is not adapted when the filtration doesn't satisfy the usual conditions. Another source of detail on these things is Chapter 3 of *Introduction to Stochastic Integration* by K.L. Chung and R.J. Williams.


