Convexity of limits of harmonic measures

Abstract

It is shown that, given a point $x\in\mathbbm{R}^d$, $d\ge 2$, and open sets $U_1,...,U_k$ containing $x$, any convex combination of the harmonic measures for $x$ with respect to $U_n$, $1\le n\le k$, is the limit of a sequence of harmonic measures for $x$ with respect to open subsets $W_m$ of $U_1\cup... \cup U_k$ containing $x$. This answers a question raised in connection with Jensen measures. More generally, we prove that, for arbitrary measures on an open set $W$, the set of extremal representing measures, with respect to the cone of continuous potentials on $W$ or with respect to the cone of continuous functions on the closure of $W$ which are superharmonic $W$, is dense in the compact convex set of all representing measures. This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then shrinking these balls in a suitable manner. The results are presented simultaneously for the classical case and for the theory of Riesz potentials. Finally, a characterization of all Jensen measures and of all extremal Jensen measures is given.