This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes:
Are martingales progressively measurable? (At least up to a modification?)
The answer should be yes if it is true, as I have read (for example here), that every measurable and adapted process has a progressively measurable modification, since obviously martingales are measurable and adapted.
I ask because I want to deduce whether or not the space of locally $L^2-$bounded martingales is closed under integration with respect to Brownian motion. This seems to follow from Definition 3.1.4 p. 25, and Corollary 3.2.6 on p. 33 of Oksendal's Stochastic Differential Equations.
EDIT: The answer is almost certainly yes, since Definition 3.14 in Oksendal essentially states that: $f \in \mathcal{V}(S,T)$ if and only if: (1) $f$ is a measurable process, (2) $f$ is an adapted process, and (3) $f$ is bounded in $L^2$ on $[S,T]$. From (1) and (2) it should follow that $f$ has a progressively measurable version. Then from corollary 3.17 (Ito's Isometry) and condition (3) of Definition 3.14, it follows that the Ito integral with respect to Brownian motion over a finite interval $[S,T]$ is bounded in $L^2$, and then Corollary 3.2.6 gives us that it is a martingale, hence an $L^2$-bounded martingale.
Does this mean that if we restrict the class of integrands from "locally bounded progressively measurable processes" to "locally bounded and locally $L^2$ bounded progressively measurable processes" that the Ito integral with respect to Brownian motion is always a (locally) $L^2$-bounded martingale?
Namely we get from the second result that any integral of a locally $L^2-$bounded progressively measurable process is a martingale. Also Ito's isometry has something to do with $L^2-$bounded martingales. But anyway the space wouldn't be closed under integration w.r.t. Brownian motion if the resulting martingale wasn't itself integrable w.r.t. Brownian motion.
In what follows, we assume the existence of an underlying filtration $(\mathcal{F}_t)_{t\ge 0}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
adapted process - a process $(X_t)_{t \ge 0}$ such that $X_t : \Omega \to \mathbb{R}$ is measurable with respect to $\mathcal{F}_t$ for all $t \in [0,\infty)$.
measurable process - a process $(X_t)_{t \ge 0}$ such that the induced mapping $$X: \Omega \times [0, \infty) \to \mathbb{R}, \quad (\omega,t) \mapsto X_t(\omega)$$ is measurable (= jointly measurable in $\Omega \times [0, \infty)$????).
progressively measurable process - a process $(X_t)_{t \ge 0}$ such that, for every $t \in [0, \infty)$, the map $$[0,t] \times \Omega \to \mathbb{R}, \quad (s, \omega) \mapsto X_s(\omega)$$ is $\mathcal{B}([0,t])\times \mathcal{F}_t$ measurable.