Disintegration of dominated monotone sublinear functionals on the space of measurable functions

The article is concerned with the direct problem of disintegration of monotone sublinear func¬tional $N$ that are dominated by $\sigma$-additive measures and are continuous on increasing sequences of functions. The disintegration problem reduces to solving the nonlinear equation $N(Q(\cdot))=N(\cdot)$. We study necessary and sufficient conditions for the existence of a solution $Q$. Formulas are derived that represent $Q$ as the supremum of a family of linear operators. As an example, we prove an existence theorem for the essential (in measure) maximum of the function relative to an arbitrary $\sigma$-algebra and give a formula for its computation. The results of the article are applicable to the problems of operations research and decision making. Some of the problems are discussed in the article.