We define $$\int_{0}^Tf(X_t,t)dB_t:=\lim_{n\to \infty }\sum_{k=0}^{n-1}f(X_{t_{i-1}},t_{i-1})(B_{t_{i+1}}-B_{t_i}),$$ where the limit is taken in $L^2$, $\{t_0,...,t_n\}$ is a partition of $[0,T]$ s.t. $\max_{i=0,...,n-1}|t_{i+1}-t_i|\to 0$.
Why do we take the limit in $L^2$ ? The pointwise limit doesn't work ? (or the convergence almost surely) If no, I don't really see why... Could someone explain ?