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Capacities of Borelian Sets and the Continuity of Potentials

Published online by Cambridge University Press:  22 January 2016

Masanori Kishi
Masanori Kishi*
Affiliation:
Mathematical Institute, Nagoya University
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One of the most important problems in the potential theory is the one of capacitability, that is, whether the inner capacity of an arbitrary borelian subset B is equal to the outer capacity of B. As for the capacities induced by the Newtonian potentials and other classical potentials, Choquet [5] has shown that every borelian and, more generally, every analytic set are capacitable. He goes on as follows : first he shows that, for the Newtonian capacity f, the inequality of strong subadditivity holds, that is,

and then, using this inequality, he shows that the outer capacity f* has the analogous property to one of the outer measure, more precisely, if an increasing sequence {An} of arbitrary subsets converges to A, then f*(A) = limf*(An)- This property plays an important role in his proof.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 12 , December 1957 , pp. 195 - 219
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

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