Take an adapted process to a given non tricky filtration and dependent on two "variables", omega (event, not really a variable) and $t$ (time).
By definition adaptation means only measurability with respect to the filtration.
I am unable to prove with discrete sums and limits that the quadratic variation of the integral over $[0, t]$ of the adapted process, at a given omega, is zero.
I can't find any proof of it in books whereas plenty of them just write that the adapted process times $dt^2 = 0$ as if the formal proof (not this recipe) had been given ahead.
The proof is easy when the adapted process is continuous or bounded over $[0, t]$ but when only measurable, I am stuck.
One finds often the CLAIM of a zero quadratic variation of the drift imposed to the original brownian when proving Girsanov. But not the PROOF with discrete sums, partition steps and limits, if ever.
Any one can help ?