ИСТИНА

In 1867 Riemann for interval T introduced the Riemann integral and the notion of Riemann integrable function f on T. After that Lebesgue constructed the Lebesgue measure λ on T , and gave the famous functional description of Riemann integrable functions. Concider the factor-family R of equivalence classes of all Riemann integrable function f on T with respect to λ. Ten we get the remarkable Riemann extension R of the lattice linear space C of all continuous bounded functions f on T. Long time any functionally-analytic inter-relation between the whole families C and R were very mysterious, because nothing was known about approximation of Riemann integrable functions by means of sequences continuous functions, In this relation the remarkable Lebesgue functional description is not informative. In 1995 the author gave the other functional description of Riemann integrable functions different from Lebesgue’s one (see [2; 3.7.2, Theorem 3]). It gave to the author the opportunity to get the characterization of the Riemann extension R of C in some category of lattice linear extensions A of C. In this category the Riemann extension is characterized as some completion of C obtained by the adjunction to C some tight order boundaries of some tight countable cuts in C. The tightness is determined by some new functionally-analytic structure, called by the author the refinement [1]. The mentioned characterization is given for the general case of an arbitrary completely regular topological space T and for the family of functions on T, μ-Riemann-integrable with respect to an arbitrary positive bounded Radon measure μ with the support equal to T (see [2; 3.7]). 1. Zakharov V.K. Description of extensions of the family of continuous functions by means of order boundaries, Doklady Math. 71(2005), no. 1, 80-83. 2. Zakharov V.K., Rodionov T.V., Mikhalev A.V. Sets, Functions, Measures. Volume 2: Fundamentals of functions and measure theory. Germany, Berlin: Walter De Gruyter, 201