Let φ : $f:\Bbb R \to G $ be a measurable homomorphism, where $G$ is a Lie Group with a choice of Haar measure.
How can I prove that $\varphi$ is continuous?
When $ G=\Bbb R$ is well known that $\varphi$ is a linear function.
I also read that in general, when $G$ is locally compact, the result is also true (even in Polish Spaces, but in that case we change measurable by semi-measurable)
Help is appreciated.