# 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

MEASURES ON CERTAIN PATH SPACES OF

LIPSCHITZ FUNCTIONS WITH APPLICATIONS

TO FEYNMAN-TYPE PATH INTEGRALS

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).

**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