This dissertation develops connections between algorithmic randomness and computable analysis.;In the first part, it is shown that computable randomness can be defined robustly on all computable probability spaces, and that computable randomness is preserved by a.e. computable isomorphisms between spaces. Further applications are also given.;In the second part, a number of almost-everywhere convergence theorems are looked at using computable analysis and algorithmic randomness. These include various martingale convergence theorems and almost-everywhere differentiability theorems. General conditions are given for when the rate of convergence is computable and for when convergence takes place on the Schnorr random points. Examples are provided to show that these almost-everywhere convergence theorems characterize Schnorr randomness.
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