Analytic computable structure theory and $L^p$ spaces
Volume 244 / 2019
Fundamenta Mathematicae 244 (2019), 255-285 MSC: Primary 03D45; Secondary 03D78, 46B04. DOI: 10.4064/fm448-5-2018 Published online: 9 November 2018
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, nonatomic, 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 $p \geq 1$.