Kuhn's Equivalence Theorem for Games in Product Form

We propose an alternative to the tree representation of extensive form games. Games in product form represent information with $\sigma$-fields over a product set, and do not require an explicit description of the play temporality, as opposed to extensive form games on trees. This representation encompasses games with a continuum of actions, randomness and players, as well as games for which the play order cannot be determined in advance. We adapt and prove Kuhn's theorem-regarding equivalence between mixed and behavioral strategies under perfect recall-for games in product form with continuous action sets.