 Product of finite probability distributions

Given two finite probability distributions, assign the product of their probability masses to each ordered pair. The support is the Cartesian product of the two positive-mass supports. Distributivity gives ${\sum }_{x,y}{p}_1(x)p_2(y)=({\sum }_x{p}_1(x))({\sum }_y{p}_2(y))=1$. This is the joint distribution for independent coordinate outcomes; it is a construction from the marginals, not a claim that arbitrary jointly observed coordinates are independent.