Analytic computable structure theory and $L^p$ spaces

TL;DR

This paper investigates the computable structure theory of $L^p$ spaces, proving that for certain measure spaces, all computable presentations are isometrically equivalent, and identifying measure spaces without computable presentations.

Contribution

It establishes computable categoricity of $L^p[0,1]$ and demonstrates the existence of measure spaces lacking computable presentations despite their $L^p$ spaces being computably presentable.

Findings

$L^p[0,1]$ is computably categorical for computable $p \\geq 1$.

Every computable presentation of $L^p(\

Abstract

We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if $p \geq 1$ is a computable real, and if $\Omega$ is a nonzero, non-atomic, and separable measure space, then every computable presentation of $L^p(\Omega)$ is computably linearly isometric to the standard computable presentation of $L^p[0,1]$; in particular, $L^p[0,1]$ is computably categorical. We also show that there is a measure space $\Omega$ that does not have a computable presentation even though $L^p(\Omega)$ does for every computable real $p \geq 1$.