Abstract
The paper deals with the theory of balayage of (signed) Radon measures μ of finite energy on a locally compact space X with respect to a consistent kernel κ satisfying the domination principle. Such theory is now specified for the case where the topology on X has a countable base, while any f ∈ C0(X), a continuous function on X of compact support, can be approximated in the inductive limit topology on the space C0(X) by potentials \(\kappa \lambda :=\int \limits \kappa (\cdot ,y) d\lambda (y)\) of measures λ of finite energy. In particular, we show that then the inner balayage can always be reduced to balayage to Borel sets. In more details, for arbitrary A ⊂ X, there exists a Kσ-set A0 ⊂ A such that
μA and μ∗A denoting the inner and the outer balayage of μ to A, respectively. Furthermore, μA is now uniquely determined by the symmetry relation \(\int \limits \kappa \mu ^{A} d\lambda =\int \limits \kappa \lambda ^{A} d\mu \), λ ranging over a certain countable family of measures depending on X and κ only. As an application of these theorems, we analyze the convergence of inner and outer swept measures and their potentials for monotone families of sets. The results obtained do hold for many interesting kernels in classical and modern potential theory on \(\mathbb {R}^{n}\), \(n\geqslant 2\).
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Acknowledgements
I am deeply indebted to Bent Fuglede for reading and commenting on the manuscript, and to Krzysztof Bogdan for fruitful discussions around [4]. I express my appreciation to the anonymous referee for insightful remarks helping to improve the exposition of the paper.
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Dedicated to Professor Wolfgang L. Wendland on the occasion of his 85th birthday
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Zorii, N. On the theory of balayage on locally compact spaces. Potential Anal 59, 1727–1744 (2023). https://doi.org/10.1007/s11118-022-10024-x
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DOI: https://doi.org/10.1007/s11118-022-10024-x
Keywords
- Radon measures on locally compact spaces
- Inner and outer balayage
- Energy principle
- Consistency principle
- Domination principle
- Capacitability