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2011/2012 Mycielski-Regular Measures
Jeremiah J. Bass
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Real Anal. Exchange 37(2): 363-374 (2011/2012).


Let \(\mu\) be a Radon probability measure on the Euclidean space \(\mathbb{R}^{d}\) for \(d\geq 1\), and \(f\,:\mathbb{R}^{d}\to \mathbb{R}\) a measurable function. Given a sequence in \((\mathbb{R}^{d})^{\mathbb{N}}\), for any \(x\in\mathbb{R}^{d}\) define \(f_{n}(x)=f(x_{k})\), where \(x_{k}\) is the first among \(x_{0},\ldots, x_{n-1}\) that minimizes the distance from \(x\) to \(x_{k}\), \(0 \leq k\leq n-1\). The measures for which the sequence \((f_{n})_{n=1}^{\infty}\) converges in measure to \(f\) for almost every sequence \((x_{0},x_{1},\ldots)\) are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set \(C\) is Mycielski-regular.


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Jeremiah J. Bass. "Mycielski-Regular Measures." Real Anal. Exchange 37 (2) 363 - 374, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 15 April 2013

zbMATH: 1290.28007
MathSciNet: MR3029763

Primary: 28A02
Secondary: 60A02

Keywords: Measures , Probability

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 2 • 2011/2012
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