  L\'evy's Upward Theorem says that the conditional expectation of an
integrable random variable converges with probability one to its true value
with increasing information. In this paper, we use methods from effective
probability theory to characterise the probability one set along which
convergence to the truth occurs, and the rate at which the convergence occurs.
We work within the setting of computable probability measures defined on
computable Polish spaces and introduce a new general theory of effective
disintegrations. We use this machinery to prove our main results, which (1)
identify the points along which certain classes of effective random variables
converge to the truth in terms of certain classes of algorithmically random
points, and which further (2) identify when computable rates of convergence
exist. Our convergence results significantly generalize earlier results within
a unifying novel abstract framework, and there are no precursors of our results
on computable rates of convergence. Finally, we make a case for the importance
of our work for the foundations of Bayesian probability theory.
