A functional method on amount of entropy

The theory of information, originated by Shannon, was applied in the new subject to investigate the theory of transformation with invariant measure by Kolmogorov and his school, cf. Rokhlin [12]. Recently, Halmos [7] gave a very clarified note relative to their investigations. While, in order to achieving the channel capacity in stationary finite memory channels, cf. Feinstein [6], some important properties of the entropy (the average amount of information) of information sources in these channels were studied by Khinchin [8], Takano [13], Traregradsky [14], Breiman [2], Parthasarathy [11] and others. The basic space of information sources of the channels is the doubly infinite product set A (the messages space) of the alphabet A, which becomes a compact metric space relative to the weak product topology and in which the shift transformation is a homeomorphism on A (so-called the Bernoulli automorphism).