Ergodicity of surjective continuous endomorphism of compact abelian group

Question

Ergodic Theory with a view towards Number Theory Manfred Einsiedler and Thomas Ward Page 31 Why if $f$ is invariant then we have the following equalities in the box

Answer 1

Since $f$ is invariant, $$\sum_{\chi \in \hat{X}} c_\chi \chi = f = f\circ T = \sum_{\chi \in \hat{X}} c_\chi (\chi\circ T).$$ The coefficient of the character $\chi\circ T$ on the left is $c_{\chi\circ T}$. The coefficient of the character $\chi\circ T$ on the right is $c_\chi$. So, it must be that $c_\chi = c_{\chi\circ T}$, since we have uniqueness of decomposition into characters. Doing this argument with $\chi\circ T$ instead of $\chi$ shows that $c_{\chi \circ T} = c_{\chi \circ T \circ T} = c_{\chi \circ T^2}$. Etc.