An exact functional Radon-Nikodým theorem for Daniell integrals (Q2773395)

Let \(B\) be a lattice of real valued functions on a common domain \(X\) such that \(B\) is also a unitary algebra and let \(I\) and \(J\) be positive Daniell integrals on \(B\). The symbols \(\overline{I}\) and \(\overline{J}\) represent the respective extensions of \(I\) and \(J\) to their sets of integrable functions. Sufficient conditions are given for the existence of a Radon-Nikodým derivative for \(J\) with respect to \(I\) which in this case is a positive bounded \(I\)-integrable function \(f:X \to {\mathbb R}\) (not necessarily in \(B\)) such that \(\overline{J}(u) = \overline{I}(fu)\) for every \(I\)-integrable function \(u\). NEWLINENEWLINENEWLINEIn addition to an absolute continuity requirement, the conditions involve the \(\varepsilon\)-approximate average range of \(\overline{J}\) with respect to \(\overline{I}\) relative to a bounded \(I\)-integrable function \(u\) given by \(A_{\varepsilon}(u) = \{x \in {\mathbb R}: |\overline{J}(v) - x \overline{I}(v)|\leq \varepsilon \overline{I}(v)\) for all \(I\)-integrable \(v\) such that \( 0 \leq v \leq u \}\). The conditions are phrased as an intricate inductive exhaustion of the constant function \(1\), but they appear to reduce to a requirement that for each \(\varepsilon > 0\) and each \(I\)-integrable function \(h\) satisfying \(0 \leq h \leq 1\), there is a finite or infinite series of nonnegative integrable functions \(\sum h_n\) such that \(h = \sum h_n\); \(\overline{I}(h_n)>0\); \(\{A_{\varepsilon}(h_n)\}\) is uniformly bounded independent of \(n\), \(h\) and \(\varepsilon\); and each \(A_{\varepsilon}(h_n)\) is nonempty. NEWLINENEWLINENEWLINEThe proof is constructive, drawing on techniques of \textit{H. B. Maynard} [e.g., ``A Radon-Nikodým theorem for finitely additive bounded measures'', Pac. J. Math. 83, 401-413 (1979; Zbl 0453.28004)]. However, there is no discussion of the relationship of the result to the classical Radon-Nikodým theorem on the measure spaces determined by \(I\) and \(J\).