@MISC{Baker07anuncountable, author = {R. L. Baker and R. L. Baker}, title = {An Uncountable Family of Regular Borel measures}, year = {2007} }

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Abstract

ABSTRACT. Let c> 0 be a fixed constant. Let 0 ≤ r < s be an arbitrary pair of real numbers. Let a, b be any pair of real numbers such that | b − a | ≤ c(s − r). Define C s r to be the set of continuous real-valued functions on [r, s], and define Cr to be the set of continuous real-valued functions on [ r, +∞). Finally, consider the following sets of Lipschitz functions: Λ s r = { x ∈ Cs r | |x(v) − x(u) | ≤ c|v − u|, for all u, v ∈ [ r, s]}, (1) Λr = { x ∈ Cr | |x(v) − x(u) | ≤ c|v − u|, for all u, v ∈ [ r, +∞)}, (2) Λ s,b r,a = { x ∈ Λsr | x(r) = a, x(s) = b}, (3) Λ s r,a = { x ∈ Λ s r | x(r) = a}, (4) Λ s,b r = { x ∈ Λsr | x(s) = b}, (5) Λr,a = { x ∈ Λr | x(r) = a}. (6) 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). Key words: infinite dimensional Lebesgue measure, Lipschitz functions, Radon measures, uniform probability probability measure.