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Martin-Löf randomness and other randomness notions arising from computable tests; as well as related concepts such as Kolmogorov complexity, K-triviality, and effective Hausdorff dimension.
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Non-relativized, Computable and Schnor randomness w.r.t a measure
Riemann and Slaman have some great work classifying what reals are 1-random with respect to a measure $\mu$ relative to $\mu$. In that paper they cite Levin and Kautz (but not to refs I can find) for …
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Kurtz randomness and supermartingales with infinite *limit*
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^\om …