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 get a 'valid' randomness notion because any sufficiently generic real should qualify. However, if I had to guess I'd think this notion turns out to be equivalent to Kurtz 'randomness' (avoids all $\Pi^0_1$ null sets). Since I'm guess this is a known result and my brain is feeling super foggy I thought I would ask