Any function I can actually write down is measurable, right?
James E. Hanson

TL;DR
This paper explores the limits of measurability for complex functions defined via polynomials and set-theoretic principles, revealing independence results from ZFC and connections to large cardinal hypotheses.
Contribution
It constructs specific functions whose measurability is independent of ZFC and demonstrates how strong set-theoretic assumptions influence measurability of definable functions.
Findings
Existence of a polynomial with a function's measurability independent of ZFC
Functions with measurability linked to large cardinal hypotheses
Connections between set theory, measure theory, and definability
Abstract
In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert's tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial of degree with integer coefficients such that it is independent of (and much stronger theories) whether the function is Lebesgue measurable. We also give similarly defined with the property that the statement " is measurable for every " has large cardinal consistency strength (and in particular implies the consistency of…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Mathematical and Theoretical Analysis
