I would like to know that: Is it necessary that a stochastic process $X$ has to be adapted to some filtration $\{\mathcal{F}_t|t\in[0,T]\}$ for define the next integral?
\begin{equation} \int_0^TX_tdt \end{equation}
or it is possible to calculate this integral path-by-path even in the case when $X$ is not adapted. I know just a little about Itô's theory, and I know that in the case where the integral is respect to a Brownian motion (or martingale) the process has to be adapted.
I really appreciate any advice about where I could learn about these kind of integrals, ans their properties.
Thanks for the help. P.D. English is my second languange so my apologies for grammatical errors (I am trying to improve everyday).
No, $X$ does not need to be adapted for $\int_0^T X_t dt$ to be defined. As you said, it can be defined path-by-path. Incidentally, every stochastic process is adapted to some filtration, specifically the one generated by itself: $\mathcal F_t^X := \sigma(X_s : 0 \le s \le t)$.