## On the foundations for a measure theory and integration in two and three dimensions and a theory of delta functions over the Levi-Civita field

##### Abstract

The field of real numbers does not permit a direct representation of the (improper) delta functions used for the description of impulsive (instantaneous) or concentrated (localized) sources. Of course, within the framework of distributions, these concepts can be accounted for in a rigorous fashion, but at the expense of the intuitive interpretation. The existence of infinitely small numbers and infinitely large numbers in the Levi-Civita field allows us to have well-behaved delta functions which, when restricted to the real numbers, reduce to the Dirac delta function. Here we develop the foundations for a mathematically rigorous theory of localized and instantaneous sources that has a clear and unambiguous way of specifying a mathematically concentrated source. We use
the already existent one variable measure and integration theory on Levi-Civita field to construct the foundations of a measure and integration theory in two and three dimensions. First we construct measurable sets using sets with boundaries that can be expressed as analytic functions and we show the the resulting measure is Lebesgue-like.
In particular we prove the measurability of countable sets, the countable union of measurable sets, and the finite intersection of measurable sets. Following that we use analytic functions to construct a larger class of measurable functions, we then define the integral of a measurable function over a measurable set. We prove several propositions regarding measurable functions and the associated integration theory including that the set of measurable functions is closed under multiplication and addition, and that integration is linear.
This allows for a wide range of applications for the delta function in one, two, and three dimensions and sets the course for a more extensive study of this topic in the future.