A Mathematical Theory of Communication 



By C. E. SHANNON 



{Concluded from July 1948 issue) 



PART III: MATHEMATICAL PRELIMINARIES 



In this final installment of the paper we consider the case where the 

 signals or the messages or both are continuously variable, in contrast with 

 the discrete nature assumed until now. To a considerable extent the con- 

 tinuous case can be obtained through a limiting process from the discrete 

 case by dividing the continuum of messages and signals into a large but finite 

 number of small regions and calculating the various parameters involved on 

 a discrete basis. As the size of the regions is decreased these parameters in 

 general approach as limits the proper values for the continuous case. There 

 are, however, a few new effects that appear and also a general change of 

 emphasis in the direction of specialization of the general results to particu- 

 lar cases. 



We will not attempt, in the continuous case, to obtain our results with 

 the greatest generality, or with the extreme rigor of pure mathematics, since 

 this would involve a great deal of abstract measure theory and would ob- 

 scure the main thread of the analysis. A preUminary study, however, indi- 

 cates that the theory can be formulated in a completely axiomatic and 

 rigorous manner which includes both the continuous and discrete cases and 

 many others. The occasional liberties taken with limiting processes in the 

 present analysis can be justified in all cases of practical interest. 



18. Sets and Ensembles of Functions 



We shall have to deal in the continuous case with sets of functions and 

 ensembles of functions. A set of functions, as the name implies, is merely a 

 class or collection of functions, generally of one variable, time. It can be 

 specified by giving an explicit representation of the various functions in the 

 set, or implicitly by giving a property which functions in the set possess and 

 others do not. Some examples are: 

 1. The set of functions: 



/e(0 = sin {t + S). 



Each particular value of 6 determines a particular function in the set. 



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