4 - Algorithmic randomness and layerwise computability

Summary

Algorithmic randomness lies at the intersection between computability theory and probability theory. In order to fully explore this interaction, one naturally needs a computable version of measurable functions. While several such notions appear in the literature, most of them do not interact well with algorithmic randomness because they are only defined up to a null set. Therefore, we need a computable notion of measurable function which is well defined on algorithmically random points, and this is what layerwise computability precisely does. This article is a survey about this notion. We give the main definitions, the most important properties, and several applications of this notion. We prioritize motivating this framework and explaining its salient features.