Nonstandard Models in Measure Theory and in functional Analysis

This thesis is concerned with the study of nonstandard models in measure theory and in functional analysis. In measure theory, we define elementary numerosities, that are additive measures that take on values in a non-archimedean field and for which the measure of every singleton is 1. We have shown that, by taking the ratio with a suitable unit of measurement, from a numerosity it can be defined a non-atomic real-valued measure, and that every non-atomic measure can be obtained from a numerosity by this procedure. We then used numerosities to develop a model for the probability of infinite sequences of coin tosses coherent with the original ideas of Laplace. In functional analysis, we introduce a space of functions of nonstandard analysis with a formally finite domain, that extends both the space of distributions and the space of Young measures. Among the applications of this space of functions, we develop a continuous-in-time, discrete-in-space nonstandard formulation for a class of ill-posed forward-backward parabolic equations, and on the study of the regularity and asymptotic behaviour of its nonstandard solutions. This approach proved to be a viable alternative to the study of the vanishing viscosity limit of the solution of a pseudoparabolic regularization of the original problem.