Mathematical Proceedings of the Cambridge Philosophical Society

On the conical density properties of measures on $\mathbb{R}^n$

a1 Department of Mathematics and Statistics, P.O. Box 35 (Mad), FIN-40014, University of Jyväskylä, Finland. e-mail:

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suomala v   [Google Scholar] 


We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$. As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions.

One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$, $0\,{\lt}\,\eta\,{\lt}\,1$, and suppose that $V$ is an $m$-dimensional linear subspace of $\mathbb{R}^n$. Let $\mu$ be either the $s$-dimensional Hausdorff measure or the $s$-dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$. Then for $\mu$-almost every $x\,{\in}\,\mathbb{R}^n$, there is $\theta\,{\in}\,V\,{\cap}\, S^{n-1}$ such that \[ \liminf\limits_{r\downarrow 0}r^{-s}\mu(B(x, r)\cap H(x,\theta,\eta))=0,\] where $H\left(x,\theta,\eta\right)\,{=}\,\{{y\in\mathbb{R}^n}{\ip{\br{y-x}}{\theta}\,{\gt}\,\eta|{y-x}|\}$.

(Published Online April 26 2005)
(Received November 5 2003)
(Revised January 26 2004)


1 The research was supported by the Finnish ministry of education, the Academy of Finland, project 53292, and by the Vilho, Yrjö and Kalle Väisälä Fund.