Local boundedness and the shannon entropy

Abstract

Let f be an information function. We show that, if f is bounded on an arbitrarily small nonvanishing interval contained in (0, 1), then f ≡ S where S is Shannon's measure of entropy on a 2-event space. This answers in the affirmative some closely related questions of J. Aczél and Z. Darcózy, and at the same time provides even more evidence for the Aczél-Darcózy nonnegativity question. As corollaries, we obtain a strengthened form of Lee's theorem, an improvement to a theorem of Darcózy, and an extension to the author's previous result. Furthermore, several different proofs for completing the argument on the rationals are given, including one due to P. Erdös (private communication).