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Jan 27, 2011 at 9:00 comment added Tom Leinster +1. In fact, a couple of days ago I (re)borrowed from the library the book On Measures of Information and their Characterizations by Aczél and Daróczy, mentioned on the page Ross links to. It devotes many of its pages to... well, characterizing measures of information. For example, it discusses various known characterizations of Shannon entropy, giving theorems of the form "Shannon entropy is the unique function satisfying the following functional equations".
Jan 27, 2011 at 1:45 history answered NebulousReveal CC BY-SA 2.5