Compatible measures and merging

Two measures, mu and <(mu)over tilde>, are updated as more information arrives. If with mu-probability 1, the predictions of future events according to both measures become close, as time passes, we say that <(mu)over tilde> merges to mu. Blackwell and Dubins (1962) showed that if mu is absolutely continuous with respect to <(mu)over tilde> then <(mu)over tilde> merges to mu. Restricting the definition to prediction of near future events and to a full sequence of times yields the new notion of almost weak merging (AWM), presented here. We introduce a necessary and sufficient condition and show many cases with no absolute continuity that exhibit AWM. We show, for instance, that the fact that <(mu)over tilde> is diffused around mu implies AWM.