I'm working on Katznelson, exercice 1.16 of Chapter 1. I need to show that the $n$-th Fourier coefficients of measurable homomorphism from $\mathbb{T}$ to $T^*$ are $0$ for all, except possibly one $n$. Here $T^*$ is the group of complex numbers of modulus $1$. I think that this is the same as showing that $f(t)=\exp(2\pi i k t)$ for some integer $k$. In exercice 1.11, I have already showed that this is true if $f$ is a continuous homomorphism from $\mathbb{T}$ to $T^*$.
I appreciate any help.
For $f\in L^1(\Bbb T)$ and $h\in\Bbb R$ define the translate $\tau_h f$ by $$\tau_h f(x)=f(x+h).$$An elementary argument shows that $$\lim_{h\to0}||f-\tau_hf||_1=0$$for every $f\in L^1(\Bbb T)$ (approximate $f$ in $L^1$ by a continuous function...)
But if $f$ is a homomorphism then $$\tau_h f=f(h)f,$$so $$||f-\tau_h f||_1=|1-f(h)|\,||f||_1=|1-f(h)|.$$So $\lim_{h\to0}f(h)=1$, which says that $f$ is continuous at the origin. Since $f$ is a homomorphism it follows that $f$ is continuous.