Example

— Take X to be the unit ball $\|x\| \leq 1$ in $\mathbf{R}^3$, and let $\mathscr{H}$ be a vector space of continuous functions on X, containing the restrictions to X of the affine linear functions on $\mathbf{R}^3$ and satisfying the maximum principle, that is, for every non-constant function $h \in \mathscr{H}$, the set of points of X where h attains its supremum is contained in the sphere $S_2$. It then follows easily from Props. 5 and 7 that $\operatorname{Ch}_{\mathscr{H}}(X) = \check{S}_{\mathscr{H}}(X) = S_2$. An important example of a vector space $\mathscr{H}$ satisfying the preceding conditions is the set of functions continuous on X and harmonic in the open ball $\|x\| < 1$. For these functions, one proves that the positive measure $\mu$ of mass 1 such that $\operatorname{Supp}(\mu) \subset S_2$ and $h(x) = \int h \, d\mu$ for all $h \in \mathscr{H}$ is given, if $\|x\| < 1$, by Poisson’s formula

where $\sigma$ is the measure on $S_2$ invariant under the orthogonal group and such that $\sigma(S_2) = 1$ (Ch. VII, §3, Exer. 8).*