On infinite decodable codes

In a paper by Schutzenberger and Marcus (1959) full and completable codes were defined and investigated for the case of finite codes.The present paper compares the cases of a finite and an infinite code. We introduce the space Ω of all infinite sequences of letters and define a product probability on it. We then show that in the finite case the properties of fullness and completability are equivalent to the assertion that the probability of the message set (Section 1) is positive. In the infinite case this is no longer true, and an example is given of a full code with a message set of zero probability