Another Look at the Product Measure Problem

Abstract

Here we give a positive answer to the so-called Product Measure Problem under the relatively simple hypothesis that the measure in one of the factors is inner regular and its support with the induced Hausdorff topology is locally metrizable. No special hypothesis on the other topological measure space is required. The proof is inspired by the rather imprecise conjecture that the open sets of a topological space must satisfy some restrictions in order to support any strictly positive $\sigma $-finite measure.