Let $T>0$ and for any $t\in [0,T]$, $X_t = Y_t$ a.s., i.e., they are modifications to each other. Then, what can we say about $\int_0^T X_t dW_t$and $\int_0^T Y_t dW_t$? Are they the same (a.s.)? And what about their quadratic variations $\int_0^T X^2_tdt$ and $\int_0^T Y_t^2 dt$? My thought is they fall into an equivalence class in the definition of the integrals, since $$E\left|\int_0^T (X_t^2 - Y_t^2)dt \right| \leq \int_0^TE\left|X_t^2-Y_t^2 \right|dt=0.$$
Is it true to say so?