A simple proof that $C^{\infty}({\bf R}^n,U(1))$ does not have a Haar measure

TL;DR

This paper provides a straightforward proof demonstrating that the infinite-dimensional group of smooth functions from Euclidean space to the circle group lacks a Haar measure, highlighting limitations in measure theory for such groups.

Contribution

It offers a simple, rigorous proof that the group of smooth functions from  to U(1) does not admit a Haar measure, clarifying measure-theoretic properties of infinite-dimensional groups.

Findings

No Haar measure exists for $C^{}(\u0012^n,U(1))$

The proof is simple and accessible

Highlights limitations in measure theory for infinite-dimensional groups

Abstract

We give a simple proof that there does not exist a Haar measure on the group $C^{\infty}({\bf R}^n,U(1))$.