This is part of the martingale representation proof from Rene Schilling's Brownian Motion.
Let $M_t$ be a continuous $L^2$ martingale such that $\langle M\rangle _t=\int_0^t m^2(s,\cdot)ds$ for some $\lambda_T \otimes \mathbb{P}$-almost everywhere strictly positive process $m(s,\omega)$.
I want to show that $$W_t:= \int_0^t \frac{1}{m(s,\cdot)}dM_s$$ exists, where I define $1/m(s,\omega):=0$ is $m(s,\omega)=0$.
The definition in the book says that this stochastic integral exists if $m$ is in the family of all equivalence classes in $L^2(\mu_T \otimes \mathbb{P})$ which have a progressively measurable representative, where $\mu_T$ is the measure induced by $\langle M \rangle_t$.
We have $\mathbb{E}\int_0^t \frac{1}{m^2(s,\cdot)}d\langle M\rangle_s = \mathbb{E} \int_0^t \frac{1}{m^2(s,\cdot)}m^2(s,\cdot) ds =t$. So all we need is progressive measurability but I don't see how this follows from the assumptions. I would greatly appreciate any help.