Linear measure on plane continua of finite linear measure. (English) Zbl 0687.30009

Alexander, H.

Ark. Mat. 27, No. 2, 169-177 (1989).

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

MSC:

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

Keywords:

Hausdorff measure

References:

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