C[a,b]-Valued Measure and Some of its Properties

Abstract

Let 𝐶[𝑎, 𝑏] be the set of all real-valued continuous functions defined on a closed interval [𝑎, 𝑏]. It is a commutative Riesz algebra space with unit element 𝑒, where 𝑒(𝑥) = 1 for every 𝑥 ∈ [𝑎, 𝑏]. As in the real numbers system ℝ, we define 𝐶 ̅[𝑎, 𝑏] of the extended of 𝐶[𝑎, 𝑏]. In this paper, we shall generalize the notions of outer measure, measure, measurable sets and measurable functions from 𝐶[𝑎, 𝑏] into 𝐶 ̅[𝑎, 𝑏]. This paper is a part of our study in Henstock-Kurzweil integral of functions define on a closed interval [𝑓, 𝑔] ⊂ 𝐶[𝑎, 𝑏] which values in 𝐶 ̅[𝑎, 𝑏].