Sufficient conditions for the existence of disintegrations

Let (Q, A, P) be a probability space and R a partition of Q. A necessary and sufficient condition is given for the existence of a S-additive and measurable disintegration of P on R. It is also shown that P admits a S-additive (but not measurable) disintegration on R whenever (Q, A) is a standard space and theset {(W1, W2): W1, and W2 are in the same element of R} is coanalytic in QxQ.Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using S-additive disintegrations as conditional probabilities.