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Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\omega$ just if
$$\lim_{n \to \infty} B(X\restriction_n) = \infty$$
I'm 90% sure that you don't ge...