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

The linear lattice of Riemann μ μ -integrable functions on a completely regular space with a bounded positive Radon measure μ μ 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 c -latlineals with refinements ( ≡ c r ≡ cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ μ -Riemann extension is shown to be the result of applying this procedure to the c r μ crμ -latlineal of bounded continuous functions.