Proving that all Lebesgue measurable characters are continuous.

Question

Given that a character on $\mathbb{R}^1$ is defined as a function (complex-valued function) such that $|\phi(t)|=1$ and $\phi(s+t)=\phi(s)\phi(t)$ for all $s,t\in\mathbb{R}$, how would one go about attempting to show that all Lebesgue measurable characters are continuous. Moreover, this is supposed to be able to extend to characters on $\mathbb{R}^k$, but I don't have a formal definition for this: should I assume that it is the same definition with $s$ and $t$ being vectors in $\mathbb{R}^k$? Thanks in advance!

Answer 1

Integrate: $$ \int_{a+x}^{b+x}\phi(t)\,dt =\phi(x)\int_a^b\phi(s)\,ds, $$ for all real $a<b$. Choose $a$ and $b$ so that the integral on the right is non-zero. By dominated convergence, the integral on the left is continuous in $x$; the displayed identity now implies that $x\mapsto\phi(x)$ is continuous.