I know this works for X in S but i need to prove it for $\mathcal{P}(M)$ $\text{For } X \text{ in } \mathcal{P}(M) \text{ and } M \text{ in } \mathcal{M}^2, \text{ the process } X \cdot M \text{ is a continuous square integrable martingale with initial value zero, has quadratic variation process.}$
$(X \cdot M)_t = \int_0^t X_s^2 \, d[M]_s, \text{where } \mathcal{P}(M) \text{ denotes the progressive sigma-algebra generated by }M, \text{ and the norm } \|X\|_{M,T} \text{ is defined by}||X||_{M,T} = \left( \mathbb{E} \left[ \int_0^T X_t^2 \, d[M]_t \right] \right)^{1/2}, \text{with the condition that } \|X\|_{M,T} < \infty.$