We present the equivalence between the fuzzy core and the core under minimal assumptions. Due to the exact version of the Lyapunov convexity theorem in Banach spaces, we clarify that the additional structure of commodity spaces and preferences is unnecessary whenever the measure space of agents is "saturated". As a spin-off of the above equivalence, we obtain the coincidence of the core, the fuzzy core, and the Schmeidler's restricted core under minimal assumptions. The coincidence of the fuzzy core and the restricted core has not been articulated anywhere.