Characterization of the extension by a means of order bounds for linear lattice of bounded continuous functions generated by $\mu$-Riemann integrable functions

The extension of lattice linear space of continuous bounded functions on a completely regular space, generated by $\mu$-Riemann integrable functions on this space, is considered in the paper. To characterize this $\mu$-Riemann extension some new functionally-analytical category of $c$-latlineals with refinements ($\equiv cr$-latlineals) is used. On its base the notion of $cr$-completion of some definite type is introduced. A functionally-analytical characterization of the $\mu$-Riemann extension as some $cr_{\mu}$-completion of some definite type of the $cr_{\mu}$-latlineal of continuous bounded functions is given.