Here's the question

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Let I be the unit interval, i.e.

, and let

be the N-fold cross product of I with itself, i.e., the unit N-box. Let P be that subset of

consisting of points with positive coordinates summing to 1, i.e.

Let R be the real numbers and define the information function

by

[I have modified that to correct several typos.]
H gives the amount of information in a communication system with N alternative messages where the ith message is transmitted with probability pi. (Interesting fact: If the base of the logarithm is chosen to be 2, then the unit of information is the bit, and corresponds to the amount of information in one yes-no question.) Prove that if all the messages are transmitted with equal probability, then the amount of information is equal to log N.

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Here's what I did

Let