Closedness of sum spaces and the generalized Schrödinger problem

Some general closedness properties of sum spaces of measurable functions are established. As application one obtains existence and uniqueness results for solutions of the generalized Schrödinger problem under an integrability condition but without any topological or boundedness assumptions. They also allow to prove an interesting structural result for distributions with multivariate marginals, to prove existence results for optimal approximations in additive statistical models, and to give an extension of Kolmogorov's representation theorem for continuous functions of severalvariables. This extension implies that any locally bounded measurable function has an exact representation by a neural network with one hidden layer.