  We identify the set of extreme points and apply Choquet theory to a
normalized matrix-measure ball subject to finitely many linear side
constraints. As an application we obtain integral representation formulas for
the Herglotz class of matrix-valued functions on a finitely-connected planar
domain and associated continuous Agler decompositions for the matrix-valued
Schur class over the domain. The results give some additional insight into the
negative answer to the spectral set problem over such domains recently obtained
by Agler-Harland-Raphael and Dritschel-McCullough.
