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 $c>0$ be a fixed constant. Let $0\le r<s$ be an arbitrary pair of real numbers. Let $a,b$ be any pair of real numbers such that $\vert\,b-a\,\vert\le c(s-r)$. Define $C^s_r$ to be the set of continuous real-valued functions on $[r,s]$, and define $C_r$ to be the set of continuous real-valued functions on $[\,r,+\infty)$. Finally, consider the following sets of Lipschitz functions:
\begin{align*}\Lambda^s_r &= \{\,x\in C^s_r\;\vert\;\vert x(v)-x(u)\vert\le c\v... ...\},&(5)\cr \Lambda_{r,a}&=\{\,x\in\Lambda_r\;\vert\;x(r)=a\,\}.&(6)\end{align*}
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 $c$ 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).

Received: July 15, 2011

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

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 73
Issue: 1