Computing the exponent of a Lebesgue space

TL;DR

This paper investigates the computability of the exponent in computably presentable Lebesgue spaces, proving it is computable under certain conditions but not uniformly so in general, and explores implications for the theory of stable random variables.

Contribution

It establishes conditions under which the exponent of a Lebesgue space is computable and demonstrates limitations of uniform solutions, linking to effective theory of stable random variables.

Findings

Exponent is computable when at least 2 or finite-dimensional.

No uniform solution exists with bounds on the exponent.

Results connect to the effective theory of stable random variables.

Abstract

We consider the question as to whether the exponent of a computably presentable Lebesgue space whose dimension is at least 2 must be computable. We show this very natural conjecture is true when the exponent is at least 2 or when the space is finite-dimensional. However, we also show there is no uniform solution even when given upper and lower bounds on the exponent. The proof of this result leads to some basic results on the effective theory of stable random variables.