Mycielski-Regular Measures

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.