Working through Katznelson's An Introduction to Harmonic Analysis and have been stumped by the following problem for the past few days: Show that a measurable homomorphism of $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ into $\mathbb{C}^\times$ is actually a map into the unit circle. By previous exercises, it suffices to show that the image is compact or that the homomorphism is continuous. I found a similar exercise in Rudin's Real and Complex Analysis which asks to show that every Lebesgue measurable character on $\mathbb{R}$ is continuous. After spending a good deal of time playing around with the solution to that exercise, I realized that Rudin defines a character as a complex homomorphism having modulus 1, presupposing the result in a fundamental way.
I succeeded in showing (by copying the proof of Rudin's Theorem 9.23 and the exercise), given that a measurable homomorphism must have modulus 1, it must also be continuous and therefore given by an exponential. This is a later problem in Katznelson's book. A hint to the original problem would be appreciated!