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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.15979 |
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Table of Contents:
- Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ is of the form $ϕ(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove that any Lebesgue measurable function $ϕ:\mathbb{R} \to \mathbb{R}$ that vanishes under any $d+1$ ``difference operators'' is a polynomial of degree at most $d$. More generally, we prove the continuity of any Haar measurable polynomial map between locally compact groups, in the sense of Leibman. We deduce the above result as a direct consequence of a theorem about the automatic continuity of cocycles.