The product measure can also be constructed from the rectangle algebra by Caratheodory extension

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The page's main theorem chain defines the sigma-finite product measure through sections and iterated integrals. There is a second standard construction. Its additional input is the separate verification, carried out in the references above, that the rectangle rule ${\lambda }_0(A\times B):=\mu (A)\nu (B)$ extends consistently to a premeasure on the algebra of Finite disjoint unions of measurable rectangles form an algebra generating the product sigma-algebra. Once that premeasure has been established, Assuming countable choice, a premeasure extends through its induced outer measure extends it to the generated product sigma-algebra; the extension theorem does not itself supply the premeasure verification.

When the rectangle premeasure is sigma-finite, Assuming countable choice, the Carathéodory domain is the completion of the sigma-finite extension identifies the full Caratheodory domain with the completion of that generated extension. Uniqueness of the product measure itself is supplied separately by the main product-measure theorem later on this page, not by either cited Caratheodory result.