n_{k}\right) }-M_{t}\right\vert \geq k\right\} \right] ,$$it follows that $$d\left( \left[ M^{\left( n_{k}\right) }-M\right] _{\tau _{k\left(
n_{k}\right) }^{n_{k}}},\left[ M^{\left( n_{k}\right) }-M\right] _{\tau
_{k}^{n_{k}}}\right) \underset{k\rightarrow \infty }{\longrightarrow }0,$$which yields a contradiction with $\varepsilon \leq d\left( \left[ M^{\left(
n_{k}\right) }-M\right] _{\infty },0\right) $. Thus, $\left[ M^{\left(
n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\rightarrow }0.$ The last part of the this lemma follows immediately from the first statement. $\Box $

Using the Doob’s stopping theorem we can conclude that for $M\in \mathcal{M}_{\infty }$ and an stopping time $\tau $, that $M^{\tau }\in \mathcal{M}_{\infty },$ and therefore it follows as a corollary the following result.

\[E\[|(Mn-M)thau|\]-&gt;0=&gt;\[Mn-M\]thau-&gt;0\_in\_P\]For $\left\{ M^{\left(
n\right) }\right\} _{n\in \mathbb{N}}\subset \mathcal{M}_{\infty }$, $M\in 
\mathcal{M}_{\infty }$ and $\tau $ any stopping time holds$$M_{\tau }^{\left( n\right) }\overset{L^{1}}{\rightarrow }M_{\tau
}\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{\tau }\overset{\mathbb{P}}{\longrightarrow }0.$$

*Proof.* $\left[ \left( M^{\left( n\right) }\right) ^{\tau }-M^{\tau }\right]
_{\infty }=\left[ M^{\left( n\right) }-M\right] _{\infty }^{\tau }=\left[
M^{\left( n\right) }-M\right] _{\tau }\overset{\mathbb{P}}{\longrightarrow }0.$ $\Box $

Density processes \[Sect. Density\_Processes\]
----------------------------------------------

Given an absolutely continuous probability measure $\mathbb{Q}\ll \mathbb{P}$ in a filtered probability space, where a semimartingale with the weak predictable representation property is defined, the structure of the density process has been studied extensively by several authors; see Theorem 14.41 in He, 