I'll start with some notation: $G$ is a compact group, $H_L(f)$ is the convex hull of the set of left-translates of a function $f:G \rightarrow \mathbb{C}$, $K_f$ is its closure (which is shown to be compact). Rudin proves the existence of a left- and right-invariant Haar measure on $G$ with Kakatuni's fixed point theorem, which ends up showing that for each function $f$ there's some constant function $c \in K_f$. From this, we find that this constant is unique. This defines a function $M : C(G) \rightarrow \mathbb{C}$; we then show that this is a linear functional. To finish the proof, we invoke the Riesz representation theorem that this linear functional comes from a regular Borel probability measure on $G$. However, the Riesz representation theorem requires that $M$ by continuous, and I don't see why this is the case. We have these properties (in case it somehow comes from one of these):
$f \geq 0 \implies M(f) \geq 0$
$M(1) = 1$
And $M$ is a linear functional.
Why is $M$ continuous? Is it somehow inherited from the first two properties, or does it come from the fact that $C(G)$ is the space of continuous functions on a compact group...?
