Convexity of values of vector integrals, theorems on measurable choice and variational problems

We give an account of applications of measurable many-valued mappings and theorems on convexity of finite-dimensional vector integrals to several variational problems. Theorems on convexity are carried over to vector integrals with values in function spaces, and with the help of these we obtain a aximum principle as a ecessary and sufficient extremum condition and an existence theorem for a on-linear variational problem with operator constraints of integral equality type, similar to Monge's problem on mass displacement.