  Every Dedekind complete Riesz space X has a unique sup-completion X^{s},
which is a Dedekind complete lattice cone. This paper aims to present a
systematic study this cone by extending several known results to general
setting, proving new results and, in particular, introducing for elements of
X^{s} finite and infinite parts. This enuables us to get a satisfactory
abstract formulation of some classical results in the setting of Riesz spaces.
We prove, in pareticular, a Riesz space version of Borel-Cantelli Lemma and
present some applications to it.29*
