TITLE: Is a measurable homomorphism on a Lie group smooth?
QUESTION [7 upvotes]: 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?

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.

REPLY [6 votes]: Perhaps it is interesting to know that for group cocycles the answer is in general no (even up to coboundaries). For instance the extension $\mathbb{Z} \to \mathbb{R} \to S^1$ is described by a 2-cocycle $S^1 \times S^1 \to \mathbb{Z}$ that is measurable, but this cannot be chosen to be smooth or continuous.