  This paper introduces the expanded real numbers as an ordered subring of the
hyperreal number field that does not contain any infinitesimals, and defines
the set of all integrable functions from the real numbers to the expanded real
numbers. This allows to identify the Dirac delta with a special
hyperreal-valued function of a real variable: the Dirac delta function thus
defined is a general tool, applicable for the mathematical modeling of physical
systems in which infinitely high densities occur.
