Let $M$ be a local martingale, This means that there is a sequence of Stopping times $S_n$ $\lim_n S_n = \infty$ almost surely and $M^n_t = M_{t \wedge S^n}$ is a martingale.
Now we would like to define the integral with respect to a local martingale.
For this purpose, we define the following sets
and then
An important step left is
Here is the question
One defines $X^{(n)}_t(\omega) = X_t 1_{\{T_{(n)} \geq t\}}$ Could we have defined $X^{(n)}_t(\omega) = X_{t \wedge T_{(n)}}$ instead? It seems to me that we would still be in condition to see that
$$ I^{M^{(n)}}_t(X^{(n)}) = I^{M^{(m)}}_t(X^{(m)}), \quad 0 \leq t \leq T_n$$ Is there any reason to choose the definition $X^{(n)}_t(\omega) = X_t 1_{\{T_{(n)} \geq t\}}$ instead of $X^{(n)}_t(\omega) = X_{t \wedge T_{(n)}}$ ?
