Some remarks on conditional entropy

Using a definition of conditional entropy given by Hanen and Neveu [5, 10, 11] we discuss in this paper some properties of conditional entropy and mean entropy, in particular an integral representation of conditional entropy (§ 2), and the decomposition theorem of the KolmogorovSina¯i invariant (§ 3) (see also [6–8] and [12]). There is an essential difference between Jacob's proof of the last called theorem [6] and the proof given below. The definition of a Lebesgue space, given by Rokhlin [7–9] and [12], is not used in this paper.