  For a compact convex subset $K $ of a locally convex Hausdorff space, a
measurement on $A(K)$ is a finite family of positive elements in $A(K)$
normalized to the unit constant $1_K$, where $A(K)$ denotes the set of
continuous real affine functionals on $K$. It is proved that a compact convex
set $K$ is a Choquet simplex if and only if any pair of $2$-outcome
measurements are compatible, i.e.\ the measurements are given as the marginals
of a single measurement. This generalizes the finite-dimensional result of
[Pl\'avala M 2016 Phys.\ Rev.\ A \textbf{94}, 042108] obtained in the context
of the foundations of quantum theory.
