In Karatzas and Shreve book, Chapter 3, the authors discuss the theory of stochastic integration using a filtration $\mathcal{F}_t$ that satisfies the usual hypotheses.
At page 133 point (c) a process $X$ measurable and adapted is considered. The authors says that, under these conditions, the integral process
$$F_t(\omega)=\int_0^{t\wedge T}X_s(\omega)ds$$
($T>0$ is fixed) could not be adapted. I am struggling to find an example of such a situation. I thought that measurability and adaptability were sufficient conditions, under the usual hypotheses for the filtration, to guarantee adaptability of the integral process.