Abstract
We introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties. For instance, we give a near-martingale analogous result of the fact that the space of all bounded $L^p$-martingales, equipped with the norm $\|\cdot\|_p$, is isometric to $L^p(\mathfrak{M})$ for $p>1$. We also present Doob and Riesz decompositions for the near-submartingale and provide Gundy's decomposition for $L^1$-bounded near-martingales. In addition, the interrelation between near-martingales and the instantly independence is studied.
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