Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances.
Originally, it seems that the product of Radon (or regular Borel) measures was defined by defining the product integral, and then constructing a measure from the integral. After the realization of the importance of continuity / smoothness properties of measures beyond $\sigma$-additivity, there were constructions of product measures of $\tau$-additive measures, e.g. as appears in volume 4 of Fremlin.
As far as I can tell, these product measures are always constructed in an inner-regular manner, i.e. taking two $\tau$-additive measures inner-regular with respect to a family of sets and producing a new $\tau$-additive measure on the product that is inner-regular on a product family. Is there a naturally outer-regular version of this construction? The construction of an outer-regular Borel measure from a product integral would be better characterized as constructing a product measure on the $\sigma$-ring generated by compact $G_\delta$s, and then extending this measure to an outer-regular Borel measure.
