@Tobias Let $G/\sim$ be the set of conjugacy classes, and let $\chi_1, \dots, \chi_c$ be the irreducible characters, then consider the map from $G/\sim \to \Bbb C^c$ via $[g] \mapsto (\chi_1(g), \dots, \chi_c(g))$, extend this linearly to an isomorphism $\Bbb C^{G/\sim} \to \Bbb C^c$ (as the $\chi_i$ are linearly independent).
Then because of $\chi_i(g^{-1})=\overline{\chi_i(g)}$, this is $\Bbb Z/2\Bbb Z$-equivariant, where the action on the LHS is by inversion and on the RHS is by complex conjugation
Let $(f_n)$ be a sequence in $L^p$ such that
$$
\| f_{n+1}-f_n\|_{p}
<\frac 1{2^n} \ \forall n\in\mathbb{N}
$$
and let
$$
f: X \to [0,\infty], \
x \mapsto \sum_{n=1}^\infty| f_{n+1}(x)-f_n(x)|.
$$
Use Fatou Lemma to show that $f$ takes real values almost everywhere.
We must show the...
