Given a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F_t})_{t \geq 0}, \mathbb{P})$ and a right continuous (local) $L^2$ martingale $X$, we can define the stochastic integral $$\int H dX$$ for $$H \in \mathcal{L}^2(X) = \mathcal{L}^2(\Omega \times [0, \infty), \mathcal{P}, \mu_X)$$ with $\mathcal{P} = \sigma(\mathcal{R})$ the $\sigma$-algebra of all predictable sets generated by the system $\mathcal{R}$ of predictable rectangles and $\mu_X$ the Doléans measure with respect to $X$.
This $\sigma$-algebra $\mathcal{P}$ is, as far as I know, in general a lot smaller than the product $\sigma$-Algebra $$\mathcal{F} \otimes \mathcal{B}_{[0, \infty)}.$$ (Why) can the Doléans measure not be extended to a larger $\sigma$-algebra? Can the class of integrable processes $H$ be extended if there are additional restrictions to the integrands $X$?
Also, is there a nice representation of the set of valid integrands $X$ given $H$?
Regarding the two latter questions:
Let $X$ be continuous throughout. Allowing local martingales, the space of valid $H$ actually increases. E.g. the integral $\int X dX$ ist not always defined as a stochastic integral with respect to a martingale. But it is defined as a stochastic integral with respect to a local martingale, allowing the definition of the quadratic variation $[X]$.
We have the nice representation of integrable $H$ given by $\{ H \text{ predictable} \mid \int_0^t H^2d[X] < \infty \ \forall t\}$ (because $\int_0^t H^2d[X] = [\int_0^t HdX]$). Here $H$ can be swapped with $X$ to answer my question.