Frostman lemmas for Hausdorff measure and packing measure in a product probability space and their physical application (Q1772882)

The authors establish Frostman-type lemmas for the Hausdorff and pre-packing measures on a product probability measure space \((\Omega, {\mathcal F}, \mu) = (\Omega_1\times \Omega_2, {\mathcal F}_1\times {\mathcal F}_2, \mu_1\times\mu_2)\). Based on these results, they prove a sufficient condition for the Hausdorff and packing dimensions with respect to product measure \(\mu\) of the Cartesian product \(A\times B \in {\mathcal F}_1\times {\mathcal F}_2\) to coincide.