On the Radon-Nikodym theorem (Q1917159)

Radon-Nikodým/Lebesgue decomposition theorems are investigated from the perspective of upper integrals. Given an outer integral \(N\) on a set \(X\) and a vector sublattice \(V\) of the lattice of nonnegative functions on \(X\) such that \(N\) is additive and finite on \(V^+\) there is a natural \(L_1\) space and another upper integral \(\overline {N}\) given by \(\overline {N}(f)= \inf N(g)\), taken over \(L_1\) functions \(g\geq f\). The essentialization of an upper integral \(N\) is \(\widehat {N}(f)= \sup N(f\cdot 1_A))\), taken over measurable sets \(A\) of finite measure. For two Bourbaki integrals \(I\) and \(J\) defined on a Stonean vector sublattice of the lattice of nonnegative functions on \(X\), let \(\widehat{I}\) and \(\widehat {J}\) be the corresponding essentializations of the upper integrals determined by \(I\) and \(J\). Then there is a measurable set \(A\), unique up to negligible sets, and a nonnegative measurable function \(h\), unique almost everywhere-\(I\), such that \(X\setminus A\) is \(\widehat{I}\) negligible and \(\widehat{J} (fh)=\widehat{I} (f\cdot 1_A)\) for all \([0,\infty]\)-valued functions \(f\) on \(X\). The result also holds for Daniell integrals when restricted to sigma finite essentializations, but is not true in general. These results were previously known when \(X\) is a locally compact Hausdorff space and \(V\) is the vector lattice of compactly supported continuous functions. Another Radon-Nikodým/Lebesgue decomposition theorem is given for more general upper integrals, assuming a type of decomposability.