Generalize relative majorization to two reference distributions

Develop a generalization of relative majorization on a σ-finite measure space (X, A, μ) that uses two strictly positive reference functions q and q', defined by: f majorizes g if and only if there exists a sequence of stochastic integral operators (S_n) such that S_n f converges to g in L^1(X, μ) and S_n q ≤ q'.

Background

The paper introduces and characterizes relative majorization f ≻_q g with respect to a single strictly positive reference function q via four equivalent criteria (Lorenz curves, stochastic operators, convex functionals, and piecewise-linear tests).

Immediately after establishing these results, the authors point out a natural extension in which two reference functions q and q' are allowed, leading to a preorder defined by the existence of stochastic operators (S_n) mapping f to g while taking q to something dominated by q'. They explicitly defer developing this two-reference framework to future work.