The moment problem in $K_\sigma$-space

Hamburger's moment problem is considered for a sequence of vectors in some $K_\sigma$-space ($=\sigma$ complete vector lattice). The decomposition of $K_\sigma$-space in the band of nonuniqueness and the band in which a solution is unique is obtained. For any element in the Weyl–Hamburger vector circle, a respective solution to the moment problem is constructed. The solution is constructed by extension of a positive operator according to the modified Riesz–Kantorovich method.