The measure-theoretic aspects of entropy. II (Q1208549)

Let \(P\) be a countably additive probability measure, and \(\mu\) be any countably additive \([0,\infty]\)-valued measure, both on a \(\sigma\)- algebra \({\mathcal A}\) over a space \(\Omega\). In Part I [ibid. 40, No. 2, 215-232 (1992; Zbl 0753.60004)] we defined a \((P,\mu\) dependent) countably additive measure \(H\) on the \(\delta\)-ring \({\mathcal A}_ \mu=\{A:A \in {\mathcal A}\) \& \(\mu (A) < \infty\}\) with values in \((- \infty, \infty]\) such that in case \(\Omega \in {\mathcal A}_ \mu\), \(H(\Omega)\) is the total entropy of \(P\) relative to \(\mu\). In the present paper we show that the measure \(H\) has a countably- additive extension \(\overline H\) to a pre-ring \({\mathcal P}\), \({\mathcal A}_ \mu^{\text{loc}} \supseteq {\mathcal P} \supseteq {\mathcal A}_ \mu\), and that there is a concordance between the Lebesgue decomposition of \(\overline H\) with respect to \(\mu\) on \({\mathcal P}\), and that of \(P\) with respect to \(\mu\) on \({\mathcal A}\). When \(P \prec \prec \mu\) and \(f=dP/d \mu\), we show that \(\overline H(A)=\int_ Af(\omega)\cdot\log\{f(\omega)\cdot\mu(d\omega)\), \(\forall A\in{\mathcal P}\). We study the transformation of the \(P\), \(\mu\) relative entropy under transformations of \(\Omega\) that preserve \(P\) or \(\mu\).