I want to be sure I understand the definition correctly, say as in here:
I have grown accustomed to thinking of there being 3 classes of martingales. Those satisfying the $L^1$ condition are most common, but the nonnegative ones, and the ones for which the integral exists (most general, this one means that the positive and negative integrals are not both infinite). All 3 classes have been useful. However, when people say "martingale" in the definition of "local martingale", i.e. where the stopped process has to be a martingale, they mean $L^1$ always, or at least always unless there is further clarification? However, it is also true that the RVs within the stochastic process itself may not be $L^1$?
Are other modifications of the notion as were used for regular martingales also considered often? (For instance, adding prefixes like sub- and super-, backward martingales, discrete time...)