Given a Radón measure on each of two locally compact Hausdorff spaces we consider three product measures on the product space - the classical product measure generated by measurable rectangles as in Munroe, the product measure generated by measurable rectangles and nil sets of Bledsoe and Morse, and a Radón product measure obtained by a Riesz Representation Theorem from the product linear functional of Bourbaki. For the classical product measure we prove a Fubini Theorem for compact sets although we do not know whether compact sets are measurable, a theorem giving necessary and sufficient conditions for open sets to be measurable, and a theorem that every closed СɣϬ is measurable. We prove that all three product measures agree on compact sets and thus that the Bledsoe-Morse product measure and the Radón product measure agree on open sets. Finally, we give examples to show that certain results cannot be extended.