Another look at the product measure problem (Q541400)

With \(\tau_X,\tau_Y\) \(T_2\)-topologies on \(X\) and \(Y\) respectively, \({\mathcal M}_X,{\mathcal M}_Y\) \(\sigma\)-algebras of subsets of \(X\) and \(Y\), respectively, \(\tau_X\subset{\mathcal M}_X\), \(\tau_Y\subset{\mathcal M}_Y\), \((X,\tau_X,{\mathcal M}_X,\mu_X)\) and \((Y,\tau_Y,{\mathcal M}_Y,\mu_Y)\) are taken to be complete \(\sigma\)-finite measure spaces; such measures are called topological. \(\sigma({\mathcal D})\) is used to denote the \(\sigma\)-algebra generated by a class \({\mathcal D}\) of subsets of the given set \(X\); \({\mathcal B}_X= \sigma(\tau_X)\) and \({\mathcal B}_Y= \sigma(\tau_Y)\) are the Borel \(\sigma\)-algebras generated by these topologies. The main problem is the question whether the completion of the product measure is a topological one. This problem is examined in the extreme case in which \({\mathcal M}_X\) and \({\mathcal M}_Y\) are simply the completions of \({\mathcal B}_X\) and \({\mathcal B}_Y\) with respect to \(\mu_X\) and \(\mu_Y\). The definition of product measure is the standard product derived by Carathéodory's method from an outer measure generated by the set function \[ \mu_X\otimes \mu_Y(A\times B)= \mu_X(A) \mu_Y(B), \] where \(A\in{\mathcal M}_X\), \(B\in{\mathcal M}_Y\), \(\mu_X(A)<\infty\) and \(\mu_Y(B)< \infty\). Assuming that \(\mu_X\) and \(\mu_Y\) are \(\sigma\)-finite, the product \(\mu_X\otimes\mu_Y\) is uniquely defined on the \(\sigma\)-algebra \({\mathcal A}\), which is the completion of \({\mathcal B}_X\otimes_\sigma{\mathcal B}_Y\) or, equivalently, the completion of \({\mathcal M}_X\otimes_\sigma{\mathcal M}_Y\). Then \((X\times Y,\tau_{X\times Y},{\mathcal A},\mu_X\otimes \mu_Y)\) is again a topological measure space and the question is whether \({\mathcal A}\) coincides with \({\mathcal M}_{X\times Y}\); this question is known as the product measure problem. In this paper, a positive answer is given to the 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 \(T_2\)-topology is locally metrizable. No special hypothesis on the other topological space are imposed. 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.