The measure-theoretic aspects of entropy, part I

Abstract

Let P be a countably additive probability measure and μ be any [0, ∞]-valued countably additive measure, both on a σ-algebra U over a space ω. We define a ( P-, μ-dependent) countably additive measure H on the δ-ring U μ ={A: A∈ U and μ(A)<∞} with values in (−∞, +∞] with the property that in case ω ∈ U μ, H(ω) is the total entropy of P relative to μ. We study the Lebesque decomposition of H with respect to μ.