By Lars Tyge Nielsen
Mathematical Social Sciences 8 (1984), 1–14
A conceptual framework is established for the analysis of common knowledge and communication when information structures are represented by sigma-algebras rather than partitions. It involves construction of Boolean sigma-algebras of `generalized events,’ events which are identified only up to addition or subtraction of null sets. If all agents’ posterior expectations of a random variable are common knowledge, then they are almost surely identical. If the posterior expectations are communicated back and forth, then they converge almost surely to a consensus expectation. If the posterior distributions of a random variable are communicated back and forth, then they will almost surely converge weakly to a consensus distribution.