Much recent work in algorithmic randomness has concerned characterizations of randomness notions in terms of the almost-everywhere behavior of suitably effectivized versions of functions from analysis or probability. In this work, we examine the relationship between algorithmic randomness and Lévy's Upward Martingale Convergence Theorem, in the setting of arbitrary computable Polish spaces. We show that Schnorr randoms are precisely the points at which the conditional expectations of L^1-computable functions converge to their true value. This result has natural applications to formal epistemology and the philosophical interpretation of probability: for, the natural Bayesian interpretation of this result is that belief, in the form of an agent's best estimates of the true value of a random variable, aligns with truth in the limit, under appropriate effectiveness and randomness assumptions. We also consider other randomness notions such as Martin-Löf Randomness and density randomness. This is joint work with Simon M. Huttegger (UC Irvine) and Francesca Zaffora Blando (CMU).


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