It is shown in this paper that the product family of countably many families of perfect probability measures defined on countably generated $\sigma$-fields admits a minimal sufficient statistic if and only if each component family admits a minimal sufficient statistic. Moreover, the minimal sufficient statistic of the product family is the "product" of the minimal sufficient statistics for the component families. Examples show that the assumptions on the component families cannot be omitted.
"Minimal Sufficient Statistics for Families of Product Measures." Ann. Statist. 2 (6) 1335 - 1339, November, 1974. https://doi.org/10.1214/aos/1176342887