Generalized measures of noncompactness of sets and operators in Banach spaces

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.