Order-boundary characterization of the linear lattice of Riemann $\mu$-integrable functions as a certain completion of the linear lattice of bounded continuous functions

Abstract:

The linear lattice of Riemann $\mu$-integrable functions on a completely regular space with a bounded positive Radon measure $\mu$ is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of $c$-latlineals with refinements ($\equiv cr$-latlineals) is introduced. On this basis, the procedure of $cr$-completion of certain order-boundary type is defined. The $\mu$-Riemann extension is shown to be the result of applying this procedure to the $cr_{\mu }$-latlineal of bounded continuous functions.