We introduce the notion of type-symmetric randomized equilibrium (TSRE) by requiring those agents with the same type of characteristics to choose the same randomized choice. Such a notion provides a generic micro-foundation for the macro notion of equilibrium distribution, as used in the literature on games and economies with many agents. In particular, we show that if the space of agents is modeled by the classical Lebesgue unit interval, any Nash (resp. Walrasian) equilibrium distribution in a large game (resp. economy) is uniquely determined by one TSRE. Furthermore, we provide examples to demonstrate that this uniqueness characterization does not necessarily hold when a non-Lebesgue agent space is used.