Under certain conditions, it is shown that the invariant and almost-invariant $\sigma$-fields are equivalent if and only if the invariant $\sigma$-field is independent of an appropriate sufficient $\sigma$-field. This result helps unify work of Hall, Wijsman and Ghosh and of Pfanzagl, who dealt with the forward implication and work of Berk and Bickel, who treated the reverse implication. The conditions required are that the sufficient and invariant $\sigma$-fields be essentially disjoint and together generate the $\sigma$-field of the original data.
Robert H. Berk. "A Note on Sufficiency and Invariance." Ann. Math. Statist. 43 (2) 647 - 650, April, 1972. https://doi.org/10.1214/aoms/1177692645