Consider the processin $[0,T]$,
$Y_t=\int_0^t f(s,\omega)dW_s$,
where $\int_0^t E[f(s,\omega)^2]ds<\infty$ and $W$ is a BM. Consider also,
$X_t = \int_0^t M(s,\omega)ds$,
where $M$ is some process in $[0,T]$. The question is:
Is it the case that the process $X$ is always of finite variation in $[0,T]$? Hence, is the covariation $<X,Y>_T$ always $0$?