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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

##### MSC:
 30C35 General theory of conformal mappings 28A75 Length, area, volume, other geometric measure theory
##### Keywords:
Hausdorff measure
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##### References:
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