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Let the tentative evaluation be x y. Then the function x y approaches (as T ! ∞) a constant for almost all x y which are in the high probability region corresponding to the system: ; ; ; ; ; y ! v,Px; y x and we may also write ; y ! ZZ ; x ZZ since ; Px y x y dx dy ; Px y dx dy = 1 : This establishes the desired result. 9 It measures how undesirable it is (according to our fidelity criterion) to receive y when x is transmitted. The general result given above can be restated as follows: Any reasonable evaluation can be represented as an average of a distance function over the set of messages and recovered messages x and y weighted according to the probability Px y of getting the pair in question, provided the duration T of the messages be taken sufficiently large.

Let us suppose that the power is limited to S not at every instant of time, but only at the sample points. The maximum entropy of the transmitted ensemble under these weakened conditions is certainly greater than or equal to that under the original conditions. This altered problem can be solved easily. The maximum entropy p occursp if the different samples are independent and have a distribution function which is constant from , S to + S. The entropy can be calculated as W log4S : The received signal will then have an entropy less than  W log4S + 2 eN 1 + S with !

5; “On Causality Statistics and Probability,” Journal of Mathematics and Physics, v. XIII, No. 1, 1934; N. Wiener, “The Ergodic Theorem,” Duke Mathematical Journal, v. 5, 1939. 33 Just as we may perform various operations on numbers or functions to obtain new numbers or functions, we can perform operations on ensembles to obtain new ensembles. Suppose, for example, we have an ensemble of functions f t  and an operator T which gives for each function f t  a resulting function g t : g t  = T f t  : Probability measure is defined for the set g t  by means of that for the set f t .