Optimal constrained selection of a measurable subset

Necessary and sufficient conditions are given for a class of optimization problems involving optimal selection of a measurable subset from a given measure space subject to set function inequality constraints. Results are developed firstly for the case where the set functions involved possess a differentiability property and secondly where a type of convexity is present. These results are then used to develop numerical methods. It is shown that in a special case the optimal set can be obtained via solution of a fixed point problem in Euclidean space.