Proofs of the Riesz–Markov–Kakutani representation theorem

My favourite proof is due to Hartig:

D. G. Hartig, The Riesz representation theorem revisited, American Mathematical Monthly, 90(4), 277–280.

Hartig claims that his proof is ...category-theoretic but having unwrapped the details I must say it is really functional-analytic. The idea is as follows.

• Step 1. We can easily prove this theorem for extremely disconnected compact Hausdorff spaces. Indeed, everything can be reduced there to messing around with indicator functions of clopen subsets.

• Step 2. We employ the Stone–Čech compactification of a discrete space. One can use the Banach–Alaoglu theorem to show that every completely regular space admits the Stone–Čech compactification. The Stone–Čech compactification of a discrete space is extremely disconnected.

• Step 3. Now, we prove that every compact space $X$ is a continuous image of an extremely disconnected space. Indeed, give $X$ the discrete topology. Call this space $X_d$ and extend the identity map $\iota \colon X_d \to X$ to a continuous map from $\beta X_d$ to $X$. Then, transfer the Riesz theorem to $X$ via the adjoint map to $f\mapsto f\circ (\beta\iota)$ ($f\in C(X)$).