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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
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[1] Alexander, H., Remark on a lemma of Pommerenke,Bull. London Math. Soc. 20 (1988), 327–328. · Zbl 0664.30032 · doi:10.1112/blms/20.4.327
[2] Besicovitch, A. S., On the fundamental geometric properties of linearly measurable plane sets(II), Math. Ann. 116 (1938), 296–329. · Zbl 0018.11302 · doi:10.1007/BF01448943
[3] Besicovitch, A. S., On the fundamental geometric properties of linearly measurable plane sets(III), Math. Ann. 116 (1939), 349–357. · Zbl 0020.01003 · doi:10.1007/BF01597361
[4] Moore, R. L., Concerning triods in the plane and the junction points of plane continua,Proc. Nat. Acad. Sci. U.S.A.14 (1928), 85–88. · JFM 54.0630.03 · doi:10.1073/pnas.14.1.85
[5] Pommerenke, Chr.,Univalent functions, Vandenhoeck and Ruprecht, Göttingen, 1975.
[6] Rado, T. andReichelderfer, P. V.,Continuous transformations in analysis, Grundlehren d. Math. Wiss.75, Springer-Verlag, Berlin, 1955.
[7] Rudin, W., personal communication.
[8] Falconer, K. J.,The geometry of fractal sets, Cambridge Univ. Press, 1985. · Zbl 0587.28004
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