When does randomness come from randomness?

TL;DR

This paper investigates the preservation of different notions of randomness under measure-preserving transformations, showing that Martin-Löf randomness is the largest class preserved, unlike Schnorr randomness.

Contribution

It demonstrates that measure-preserving transformations preserve Martin-Löf randomness but not Schnorr randomness, clarifying the boundaries of randomness preservation.

Findings

Martin-Löf randomness is preserved under measure-preserving transformations.

Schnorr randomness is not necessarily preserved.

The set of Martin-Löf randoms is maximal for this property.

Abstract

A result of Shen says that if $F\colon2^{\mathbb{N}}\rightarrow2^{\mathbb{N}}$ is an almost-everywhere computable, measure-preserving transformation, and $y\in2^{\mathbb{N}}$ is Martin-L\"of random, then there is a Martin-L\"of random $x\in2^{\mathbb{N}}$ such that $F(x)=y$. Answering a question of Bienvenu and Porter, we show that this property holds for computable randomness, but not Schnorr randomness. These results, combined with other known results, imply that the set of Martin-L\"of randoms is the largest subset of $2^{\mathbb{N}}$ satisfying this property and also satisfying randomness preservation: if $F\colon2^{\mathbb{N}}\rightarrow2^{\mathbb{N}}$ is an almost-everywhere computable, measure-preserving map, and if $x\in2^{\mathbb{N}}$ is random, then $F(x)$ is random.