Why the following stochastic inequality is valid

Question

Consider a Banach space-valued dyadic martingale $M_n$ on $\Omega=\{-1,1\}^{\mathbb N_*}$ associated with to $\mathcal{A}=\sigma(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$, where $\epsilon_n:\Omega\to \{-1,1\}$ denotes the $n$-th coordinate. Also, let us write martingale difference $dM_n:=M_n-M_{n-1}$. Therefore, by Representation of Paley–Walsh martingales it follows that there exists a $\mathcal{A}_{n-1}$-measurable function $\delta_{n-1}$ such that $$ dM_n=\delta_{n-1}\epsilon_n. $$ In a paper, it claim that for any $q\in (2,\infty)$, the following inequality is valid $$ \int_{\Omega\times \Omega}|M_{n-1}(\omega)+\epsilon_n(\omega')\delta_{n-1}(\omega)|^q dP(\omega')dP(\omega)\leq \int_{\Omega}|M_n(\omega)|^qdP(\omega) $$ Why ?