We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ︁ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ︁ and setsK in the context of E‐valued Bochner integrable functions defined on (...) a finite measure space. (shrink)

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ and setsK in the context of E-valued Bochner integrable functions defined on (...) a finite measure space. (shrink)

We introduce and study several nonstandard representations of Banach-valued operators defined on the space of Bochner integrable functions. They will be less restrictive than the usual standard representation. In particular, without any hypothesis, we shall find a representation whose kernel belongs to a space of “extended Bochner integrable functions”, introduced by Zimmer by using Loeb measures.