The Kaplan Extension of the Ring and Banach Algebra of Continuous Functions as a Divisible Hull

A new characterization of Riemann-integrable functions

“…This condition on S is very restrictive. This is why another class of functions on T was introduced in [3][4][5]. A function f : T → R is called S-uniform if for each ε > 0 there exists a finite covering σ ≡ (S i ∈ S | i ∈ I) of the set T such that the oscillation ω(f, S i ) ≡ sup(|f (s) − f (t)| | s, t ∈ S i ) of the function f over each set S i is less than ε.…”

“…In this paper, we give another description of Riemann-integrable functions with the help of a new class of uniform functions, introduced in [3][4][5] (see Corollary 3 to Theorem 3). This description allows us to uncover the "countable" nature of the relation between the above-mentioned two spaces (see Proposition 2).…”