Capacities in Complex Analysis pp 81-98 | Cite as
Gamma Capacity
Chapter
Abstract
Definition (the Choquet Integral). Assume that f is a non-negative function and c a capacity. Then \(\int {f{d_C}} \) is defined by
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$$\int {f{d_c} = \int\limits_0^\infty {c\left( {\left\{ {x;f\left( x \right) > s} \right\}} \right)ds} } $$
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Notes and references
- Theorem IX:2 is due to F. Topsøe, On construction of measures. Københavns Universitet, Mat. Inst. Preprint series 1974:27.Google Scholar
- Corollary IX:1 is due to G. Choquet, Theory of capacities. Ann. Inst. Fourier 5 (1953-54).Google Scholar
- The notation of swarm is closely related to that of “noyau capacitaire regulier” as defined in C. Dellacherie, Ensembles analytiques. Capacités. Mesures de Hausdorff. Springer LNM. 295 (1972).Google Scholar
- Example 1 is due to V. Šeinow (see Ronkins book below). Example 2 is due to C.O. Kiselman. Manuscript. Uppsala 1973.Google Scholar
- The gamma capacity was introduced in L.I. Ronkin, Introduction to the theory of entire functions of several variables. Amer. Math. Soc. Providence. R.I. 1974, and the modified gamma capacity is in S.Ju. Favorov, On capacity characterizations of sets in ₵n. Charkov 1974 (Russian). The remark by Remmert, used in the proof of Theorem IX:9 is in R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133 (1957).Google Scholar
- The set functions γn and Γn has been used in connection with removable singularity sets; cf. U. Cegrell, Removable singularity sets for analytic functions having modulus with bounded Laplace mass. Proc. Amer. Math. Soc. Vol. 88 (1983).Google Scholar
- P. Järvi, Removable singularities for Hp-functions. Proc. Amer. Math. Soc. Vol. 86 (1982).Google Scholar
- J. Riihentaus, An extension theorem for meromorphic functions of several variables. Ann. Acad. Sc. Fenn. Sér. AI. Vol. 4 (1978/79).Google Scholar
- Some of the material of this section has been published in seminaire Pierre Lelong-Henri Skoda (Analyse) 1978/79. Springer LNM 822. 1980.Google Scholar
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© Springer Fachmedien Wiesbaden 1988