pens our interest. In a scale of information interest, unexpected facts are what give information its highest scores. In contrast, there is decreasing interest in hearing or reading again the same news, except for the purpose of memorizing the information details. Jokes provide a good example here. The real information hidden in the joke is not the story but the conclusion or punch line. The joke may be a good one for various complex reasons, but an important common element is that the meaning of the punch line must be completely unexpected. A joke for which everyone in the audience already knows the conclusion is of little interest. This observation leads to a sense that the interest of information is the highest when it is fully unexpected to the recipient. The intrinsic importance of information is thus intimately associated with some degree of uncertainty. If an event is very likely, we gain little information in learning that it finally occurred. The converse is also true: unlikely or surprising events are those containing the highest information potential. Thus, information and event probabilities are connected, which bring us on track towards a measure definition. Consider next a practical example. A lotto surprise Assume that I play the lotto, and the information I am looking after is the lotto's results. The official information finally comes as the following message: x = The winning combination is 7, 13, 17, 18. The meaningful part of this information is the four-number set. After I compare these numbers with the ones printed on my ticket, the information reduces to either X = I have won, or X = I have lost. Concisely expressed, the two possible information events are x = W or x = L, respectively, with X = {W, L} representing the event set. To each of these events (x = W or X = L) is associated a probability, noted p(x). For simplicity, we shall note p(x = W) = p(W) and p(x = L) = p L . To recall from Chapter 1, a probability is a positive number taking any value between 0 (impossible event) and 1 (abso