Linear measure on plane continua of finite linear measure
type: Item
instance of: scholarly article
title: Linear measure on plane continua of finite linear measure
zbMATH Open document ID: 0687.30009
DOI: 10.1007/BF02386369
published in: Arkiv för Matematik
publication date: 1989-01-01T00:00:00Z
review text: 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 \[ 2\int_{B}g d\Lambda =\sum_{j}\int^{2\pi}_{0}g(f_ j(e^{i\theta}))| f'(e^{\quad i\theta})| d\theta. \]
reviewed by: Christian Pommerenke
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 4125741
zbMATH Keywords: Hausdorff measure
author: Herbert J. Alexander
MaRDI profile type: Publication
cites work: Remark On a Lemma of Pommerenke
cites work: On the fundamental geometrical properties of linearly measurable plane sets of points. II
cites work: On the fundamental geometrical properties of linearly measurable plane sets of points. III
cites work: Q4052427
cites work: Q3227685
cites work: Q3714302
Recommended article: Continua of Finite Linear Measure I
Recommended article: Continua of Finite Linear Measure, II
Recommended article: Q4028232
Recommended article: Q5601863
Recommended article: Q3785382
Recommended article: Q4654986
Recommended article: Continuity of universally measurable linear maps
Recommended article: Spaces of \(\sigma \)-finite linear measure
Recommended article: Hausdorff measure of linear Cantor set
Recommended article: Uniform measures on linear spaces
description: scientific article; zbMATH DE number 4125741