# IJPAM: Volume 73, No. 1 (2011)

AN UNCOUNTABLE FAMILY OF REGULAR BOREL
MEASURES ON CERTAIN PATH SPACES OF
LIPSCHITZ FUNCTIONS WITH APPLICATIONS
TO FEYNMAN-TYPE PATH INTEGRALS

R.L. Baker
Department of Mathematics
University of Iowa
Iowa City, Iowa 52242, USA

Abstract. Let be a fixed constant. Let be an arbitrary pair of real numbers. Let be any pair of real numbers such that . Define to be the set of continuous real-valued functions on , and define to be the set of continuous real-valued functions on . Finally, consider the following sets of Lipschitz functions:

We present a general method of constructing an uncountable family of regular Borel measures on each of the sets (1), (2), and an uncountable family of regular Borel probability measures on each of the sets (3)-(6). Using this method, we give a definition of Lebesgue measure on the sets (1) and (2), and a definition of the uniform probability measure on each of the sets (3)-(6). By interpreting as the speed of light, we then use Lebesgue measure on the sets (1), (2) and the uniform probability measure on the sets (3)-(6) to rigorously define versions of the relativistic Feynman integral and the relativistic Wiener integral on the sets of relativistic paths (1)-(6).

AMS Subject Classification: 26A99, 28C05, 28C15, 28C20, 60B05, 81S40

Key Words and Phrases: infinite dimensional Lebesgue measure, Lipschitz functions, Radon measures, relativistic Feynman integral, relativistic Wiener integral, uniform probability measure