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Linear measure on plane continua of finite linear measure. (English) Zbl 0687.30009
Let B be a plane continuum of finite linear Hausdorff measure \(\Lambda\) (B). Then B has at most countably many complementary domains \(V_ j\). These are simply connected. Let \(f_ j\) map the unit disk conformally onto \(V_ j\). The author shows that \[ 2\Lambda (b)=\sum_{j}\int^{2\pi}_{0}| f'(e^{i\theta})| d\theta. \] More generally, if g is a bounded Borel function defined on B then \[ 2\int_{B}g d\Lambda =\sum_{j}\int^{2\pi}_{0}g(f_ j(e^{i\theta}))| f'(e^{\quad i\theta})| d\theta. \]
Reviewer: Ch.Pommerenke

30C35 General theory of conformal mappings
28A75 Length, area, volume, other geometric measure theory
Full Text: DOI
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