Every state on interval effect algebra is integral

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel sigma-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying Riesz decomposition property or for every many valued (MV)-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra E can be extended to a state on E. Our method represents also every state on the set of effect operators of a Hilbert space as an integral. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3467463]