Nonstandard Point Counting for Algebraic Varieties

For varieties over a field k, we define motivic measures with values in ordered fields, using point counting over finite fields and taking the limits with respect to ultrafilters. Some properties and problems related to such measures are discussed. We define similar measures for algebraic dynamical systems by counting periodic points, and explain how they can be used to prove certain statements, such as the non-rationality of a motivic zeta function.