Properties of the "Entropy" H_λ(ξ) and the "Average Conditional Entropy" H_λ(ξ|η)

In [2], [3] and [6], the information of a random variable ξ with respect to another random variable η is defined as [numerical formula] which is a generalization of Shannon's definition of information. This quantity plays a fundamental role in information theory. The quantity I(ξ, ξ), which is called "the entropy of ξ", is often used. But, when ξ is a continuous random variable, I(ξ, ξ) becomes infinite, and so, sometimes, it is inconvenient to use I(ξ, ξ) as "the entropy of ξ". In [7], the author introduced a new quantity L(P_1, P_2, μ) and, using this, defined the entropy H(P_ξ;λ)=H_λ(ξ) of P_ξ with respect to λ. This definition of H_λ(ξ) is more general than that of "entropy" in [3] and serves as both Shannon's entropy and Wiener's one. The aims of this paper are (i) to define the "average conditional entropy" H_λ(ξ|η) (Section 3), (ii) to investigate various properties of H_λ(ξ) and H_λ(ξ|η) (Sections 4 and 5) and (iii) to consider the relations among H_λ(ξ), H_λ(ξ|η), I(ξ, η), etc, and to derive analogous formulas to usual ones