It is shown that all the possible cases can arise in the mixture problem with respect to perfectness of probability measures. A characterization of perfectness is obtained through properties of a countably generated sub-$\sigma$-algebra given which there is a regular conditional probability. Perfectness of a perfect mixture of perfect measures is characterized.
"Perfect Mixtures of Perfect Measures." Ann. Probab. 7 (3) 444 - 452, June, 1979. https://doi.org/10.1214/aop/1176995045