Describing limits of integrable functions as grid functions of nonstandard analysis

Abstract

In functional analysis, there are different notions of limit for a bounded sequence of L^1 functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of L^1 functions can be described in terms of its weak-star limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson’s nonstandard analysis, we show that for every bounded sequence {z_n}_{n in mathbb {N}} of L^1 functions there exists a function of a hyperfinite domain (i.e. a grid function) that represents both the weak-star and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward–backward parabolic equation.