Compatible measures and merging

Two measures, μ and &mutilde;, are updated as more information arrives. If with μ-probability 1, the predictions of future events according to both measures become close, as time passes, we say that &mutilde; merges to μ. Blackwell and Dubins showed that if μ is absolutely continuous with respect to &mutilde; then &mutilde; merges to μ. 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 &mutilde; is diffused around μ implies AWM.