This paper is a survey on decompositions of \(\sigma\)-additive measures and convergence theorems for measures taking values in \(l\)-groups. The contribution of the authors lies mainly on new results relating the Stone extension of a measure to its Yosida-Hewitt decomposition. In order to obtain these results, the authors employ the \(\sigma\)-ideal principle and the Stone Isomorphism technique rather than the construction of the Rickart integral.