Upper Porosity With Respect to Measures

Abstract

For subsets of a separable metric space $X$ we introduce the notion of upper porosity with respect to a~Borel regular probabilistic measure $\mu$ on $X$ (called $\mu$-upper porosity) that generalizes the concept of upper porosity of the measure $\mu$. We explore several natural definitions and further provide a definition of even more general type of $\mu$-upper porosity given by suitable porosity functions. As the main consequence of achieved results concerning general $\mu$-upper porosities we get that every $\sigma$-$\mu$-upper porous set can be decomposed to a $\sigma$-strongly upper porous set and a $\mu$-null set.