调和测度极限的凸性
Convexity of limits of harmonic measures
摘要：
证明了给定点$x\in\mathbbm{R}^d$，$d\ge 2$和包含$x$的开集$U_1,...,U_k$，$x$的调和测度关于$U_n$，$1\le n\le k$的任何凸组合，是$x$的调和测度序列关于包含$x$的$U_1\cup... \cup U_k$的开子集$W_m$的极限.这回答了与詹森措施有关的一个问题。更一般地，我们证明了，对于任意的措施在一个开放的集合$W$，极值表示措施的集合，相对于锥的连续潜力$W$或相对于锥的连续函数的封闭$W$，这是超调和$W$，是稠密的紧凸集的所有表示措施。这是通过在成对不相交且相对于它们的相互距离非常小的球的并集上进行平衡，然后以适当的方式收缩这些球来实现的。同时提出的结果为经典的情况下，理论的Riesz潜力。最后给出了所有Jensen测度和极值Jensen测度的一个刻画.
英文摘要：
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.
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