We show that an effect tribe of fuzzy sets T⊆[0,1]X with the property that every f∈T is B0(T)-measurable, where B0(T) is the family of subsets of X whose characteristic functions are central elements in T, is a tribe. Moreover, a monotone σ-complete effect algebra with RDP with a Loomis-Sikorski representation (X,T,h), where the tribe T has the property that every f∈T is B0(T)-measurable, is a σ-MV-algebra.