The ergodic decomposition defined by actions of amenable groups, part II: more about the role played by the ergodic invariant probability measures in the decomposition

Our purpose here is to continue the study of the ergodic decomposition for actions defined by amenable groups, started in [R. Zaharopol, Colloq. Math. 165 (2021)]. We consider the set $\Gamma ^{(w)}_{\alpha \mathrm {cpie}}$ defined in the above-mentioned paper, and we prove that it is Borel measurable and of maximal probability.