Representation of states on effect-tribes and effect algebras by integrals

We describe σ -additive states on effect-tribes by integrals. Effect-tribes are monotone σ -complete effect algebras of functions where operations are defined pointwise. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every σ -convex combination of extremal states on a monotone σ -complete effect algebra is a Jauch–Piron state.