  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.

Analysis of PDEs