Continuous functions on multipolar sets

Abstract:

Let $\Omega = {\Omega _1} \times \cdots \times {\Omega _n}(n > 1)$ be a product of $n$ Brelot harmonic spaces each of which has a bounded potential, and let $K$ be a compact subset of $\Omega$. Then, $K$ is an $n$-polar set with the property that every $i$-section $(1 \leqslant i < n)$ of $K$ through any point in $\Omega$ is $(n - i)$ polar if and only if every positive continuous function on $K$ can be extended to a continuous potential on $\Omega$. Further, it has been shown that if $f$ is a nonnegative continuus function on $\Omega$ with compact support, then $MRf$, the multireduced function of $f$ over $\Omega$, is also a continuous function on $\Omega$.