Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?




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**Edit:** My original question said "measurable function" instead of the more accurate "measurable homomorphism." Marc Palm and other people answered both questions very nicely: 


* there are obviously non-smooth measurable functions on Lie groups, and
* all measurable homomorphisms on Lie groups are smooth.



Is a measurable homomorphism on a Lie group smooth?
No, take a proper closed set $X$ and define $f|X = 1$ and $f|G-X = g \neq 1$.


If you consider $f$ to be a group homomorphism, then the answer is yes in general.


