A Radon-Nikodym derivative for almost subadditive set functions

In classical measure theory, the Radon-Nikodym theorem states in a concise condition, namely domination, how a measure can be factorized by another (bounded) measure through a density function. Several approaches have been undertaken to see under which conditions an exact factorization can be obtained with set functions that are not σ-additive (for instance finitely additive set functions or submeasures). We provide a Radon-Nikodym type theorem with respect to a measure for almost subadditive set functions of bounded sum. The necessary and sufficient condition to guarantee a one-sided Radon-Nikodym derivative remains the standard domination condition for measures