## Real Analysis Exchange

### Mycielski-Regular Measures

Jeremiah J. Bass

#### 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

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

#### 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.