I am currently reading the first chapter on the general theory of stochastic processes in "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev. On page 41 they state the following:
4.13 Lemma: c) Let $M$, $N$ be two orthogonal local martingales. For all stopping times $S$, $T$, the stopped local martingales $M^S$ and $N^T$ are orthogonal.
The definition of orthogonality they provide on page 40 is the following:
Definition: Two local martingales $M$ and $N$ are called orthogonal if their product $MN$ is a local martingale.
The proof goes as follows: They reduce to the case where $T=\infty$. Then by localization, we can find a sequence of stopping times $(T_n)$ such that $M^{T_n}, N^{T_n}, (MN)^{T_n}$ are all martingales. Then for a stopping time $R$, they write
\begin{align} E[(M^S N)^{T_n}_R] &= E[(MN)^{T_n}_{R \land S}] + E[M^{T_n}_S (N^{T_n}_R - N^{T_n}_S)1_{S < R}] \\ &=E[M_0 N_0] + E[M^{T_n}_S 1_{S < R} E[N^{T_n}_R-N^{T_n}_S |\mathcal{F}_S] = E[M_0 N_0] \end{align}
where in the second step they use the stopping theorem and another characterisation of being a martingale (1.44 Lemma, I can provide further details if needed).
But the only part I do not understand, is why in the first expectation, $(M^S N)^{T_n}_R$ is even integrable? Since we only know integrability of the processes $M^{T_n}, N^{T_n}, (MN)^{T_n}$. Does anyone know what the reasoning behind the integrability is?
Thanks a lot in advance!