An introduction to analysis on linearly ordered extensions of $\mathbb R$

Emanuele Bottazzi

Analysis on linearly ordered extensions of $\mathbb R$ holds the promise to be relevant for many applications. An example is the mathematical description of physical phenomena via partial differential equations: many non-Archimedean formulations enable the representation of distributions and Young measures as pointwise functions.

We provide a gentle introduction to analysis on linearly ordered field extensions of the real numbers. Recall that such extensions are non-Archimedean: due to this feature, continuous and differentiable functions have very different properties than the ones of their real counterparts. For instance, continuous functions do not have the intermediate value property and the space of solutions of the ODE $f’ = 0$ is infinite-dimensional. Different extensions handle such problematic features in different ways: for instance, in hyperreal fields of Robinson’s nonstandard analysis, it is possible to work only with internal objects, namely sets, functions, relations, etc. that satisfy the same first order properties as their real counterparts.

We focus on real closed field extensions of the real numbers that are complete in the order topology and with an Archimedean skeleton group. On such fields it is not possible to define internal objects. Instead, there is an ongoing research for the minimal hypotheses that entail the usual theorems of calculus. We review some recent works by Shamseddine et al. towards this goal.

Similarly, non-Archimedean measure theory on these fields faces significant obstacles. We review the uniform measure defined by Berz and Shamseddine and discuss some of its limitations. We argue that an alternative approach based on a real-valued measure inspired by the Loeb measure of nonstandard analysis might overcome some of those limitations and enable further applications.

Throughout the presentation we discuss non-Archimedean representatives of the Dirac distribution. Shamseddine et al. proposed a polynomial representative that allowed to calculate directly the Green function of some simple PDEs. By using the real-valued measure, we were able to establish a limited integration by parts formula analogous to that of the distributional derivative.