If you're talking about convergence under the $L2$-norm, I think the answer is probably no.
For example, take $M_t = 1$, and $g(t)$ as a step function. The "derivative" of $g(t)$ is a sum of delta functions, which are not in $L2$ (??).
If you're talking about convergence under the $L2$-norm, I think the answer is probably no.
For example, take $M_t = 1$, and $g(t)$ as a step function. The "derivative" of $g(t)$ is a sum of delta functions, which are not in $L2$ (??).