Real Analysis Exchange

Mycielski-Regular Measures

Jeremiah J. Bass

Full-text: Open access

Abstract

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.

Article information

Source
Real Anal. Exchange, Volume 37, Number 2 (2011), 363-374.

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1366030631

Mathematical Reviews number (MathSciNet)
MR3029763

Zentralblatt MATH identifier
1290.28007

Subjects
Primary: 28A02
Secondary: 60A02

Keywords
measures probability

Citation

Bass, Jeremiah J. Mycielski-Regular Measures. Real Anal. Exchange 37 (2011), no. 2, 363--374. https://projecteuclid.org/euclid.rae/1366030631


Export citation

References

  • Falconer, K. J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 1985.
  • Fremlin, David, Problem GO, http://www.essex.ac.uk/maths/people/fremlin/problems.htm
  • Fremlin, David, Measure Theory, Vol 2, Torres Fremlin, Colchester, 2001
  • Mycielski, Jan, Learning Theorems, Real Anal. Exchange, to appear.