M: Revealed preference, rational inattention, and costly information acquisition. Am Econ Rev 2015

Proof. Let 2 be an information structure that is consistent with PA 2 P. First, we list in order all distinct posteriors i 2 () for 1 i I = j()j. Given that is consistent with PA, there exists a corresponding optimal choice strategy CA: () ! (A) such that the information structure and choice functions match the data, PA(aj!) = IX i=1 (ij!)CA(aji): We also list in order all possible posteriors associated with the corresponding revealed in-formation structure, j 2 (A), 1 j jj, and identify all chosen actions that are associated with posterior j as F j, Fj fa 2 Aj aA = jg: The garbling matrix bij sets the probability of j 2 given i 2 () as the probability of all choices associated with actions a 2 Fj bij = X a2 Fj CA(aji): Note that this is indeed a j()j jj stochastic matrix B 0 withPJj=1 bij = 1 all i. Given j 2 () and! 2, note that, IX i=1 bij(ij!) = IX i=