Attainable values for upper porosities of measures (Q1852354)

To a measure \(\mu\) over \(\mathbb R^{n}\) one associates the upper and lower porosity, denoted \(\overline{p}(\mu)\), respectively \(\underline{p}(\mu)\), in terms of the upper, respectively lower porosity of all subsets in \(\mathbb R^{n}\) of positive \(\mu\)-measure. The main result of the paper is that upper porosity of a Radon probability measure on \(\mathbb R^{n}\) is either \(0\) or \(1/2\). The authors introduce a second definition of upper porosity, denoted \(\overline{\text{por}}(\mu)\), which in the case of measures satisfying the so-called doubling condition is equivalent to the first one. It is proved that for a Radon probability measure, \(\overline{\text{por}}(\mu)\) is \(0, 1/2\) or \(1\).