The following question pertains to Wengenroth's textbook "Wahrscheinlichkeitstheorie", de Gruyter 2008 (in German).
The covariance (aka compensator) of the continuous local martingales $X, Y \in \mathcal{CM}^{\text{loc}}(\mathcal{F})$ is defined (in Theorem 9.6 on p. 181) as $$ [X, Y] := XY - X_0Y_0 - X\cdot Y - Y\cdot X $$ where $X\cdot Y$ is the Itô integral of $X$ w.r.t. the integrator $Y$: $(X \cdot Y)_t = \int_0^t X_s dY_s$, and likewise $Y\cdot X$. However, the Itô integral w.r.t. an integrator that is a local martingale has only been defined (p. 179) when the integrand is in $\overline{T}(\mathcal{F})$, i.e. when the integrand can be uniformly approximated by a step-process, where a step-process is defined (p. 176) as a stochastic process that is adapted to the filtration $\mathcal{F}$, and such that each of its paths is a step-function, i.e. a function of the form $\sum_{n = 0}^\infty a_n \mathbb{1}_{[t_n, t_{n + 1})} = \sum_{n = 0}^\infty a_n(\omega) \mathbb{1}_{[t_n(\omega), t_{n + 1}(\omega))}$, for some sequence of real numbers $a(\omega) = (a_1(\omega), a_2(\omega), \dots)$ and some sequence of non-negative real numbers $(0 = t_0(\omega), t_1(\omega), \dots)$ that increases monotonically (possibly weakly) to infinity: $t_n(\omega) \uparrow \infty$.
So the definition of covariance/compensator, given above, begs the question: is every local martingale in $\overline{T}(\mathcal{F})$? If not, then the definition makes no sense, does it? Or am I missing something?
Fix $\epsilon>0$. Define iteratively a sequence of stopping times by
$$\tau_n := \inf\{t>\tau_{n-1}; |X(t)-X(\tau_{n-1})| \geq \epsilon\}, \qquad n \in \mathbb{N},$$
$\tau_0 := 0$. Then
$$f(t,\omega) := \sum_{n \geq 1} X_{\tau_{n-1}}(\omega) 1_{[\tau_{n-1}(\omega),\tau_n(\omega))}(t)$$
satisfies $\|X-f\|_{\infty} \leq \epsilon$ (since $X$ has continuous sample paths) and $f$ is adapted.