Computable randomness and betting for computable probability spaces

TL;DR

This paper extends the concept of computable randomness to arbitrary computable probability spaces, introduces new randomness notions, and provides machine characterizations, advancing the theoretical understanding of algorithmic randomness.

Contribution

It defines computable randomness outside Cantor space, introduces endomorphism randomness, and generalizes bit-wise randomness definitions to broader spaces.

Findings

Defined computable randomness for general spaces

Introduced endomorphism randomness concept

Extended machine characterizations of randomness

Abstract

Unlike Martin-L\"of randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness---endomorphism randomness---which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-L\"of randomness. The last section contains ideas for future research.