The Riesz–Markov–Kakutani representation theorem tells us that, if $X$ is a locally-compact Hausdorff space, then monotone linear functionals $\mathcal{C}_{\mathrm{cs}}(X) \rightarrow \mathbb{R}$ are the same thing as regular measures on $X$.
I don't know about you, but the correctness of this theorem kind of messes with some broad intuitions I have about these kinds of things. In particular, the Riesz–Markov–Kakutani representation theorem violates the pseudotheorem that being finitely-linear isn't enough to establish being countably-linear in any sense of the word.
Question. Can anyone offer any intuition or justification for why it's reasonable to expect this pseudotheorem to fail here?