To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem:
Let $G$ be a compact abelian topological group with Haar measure $m$. Let $\hat G$ be the dual. The members of $\hat G$ form an orthonormal basis for $L^2(m)$.
Here I do not know about existence of Haar measure and I took it for granted. I was able to show that the members of $\hat G$ are orthonormal. How to show that these form a basis? Stone-Weierstrass(whose proof too I don't know), reads as:
Let $X$ be a compact Hausdorff space and let $A$ be a closed subalgebra of the space of complex continuous functions $\mathcal C(X,\mathbb C)$ which separates points, contains a nonzero constant function and contains the conjugate of each of its functions. Then $A$ equals $\mathcal C(X,\mathbb C)$.
Here it is easy to show everything except the fact that the subspace of $L^2(m)$ generated by the characters separate points. How to do this?