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Generalized moment problem in vector lattices. (English) Zbl 0948.44003
The classical theorem on the moment problem going back to F. Hausdorff [Math. Z. 16, 220-248 (1923; JFM 49.0193.01)] states that a given real sequence \(a_{k}\in\mathbb{R}\) is a sequence of moments of some nondecreasing function \(g\in V(0,1)\) of bounded variation, i.e., \(a_{k}=\int_{0}^{1}t^{k} dg(t)\), if and only if the sequence \(a_{n}\) is completely monotone, i.e., \(\Delta ^{n}a_{k}:=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}a_{k+j}\geq 0\). The authors generalise this result to the case where \(a_{k}\) is the sequence of elements in an ordered vector space \(V\), more precisely in \(\sigma \)-complete weakly \(\sigma \)-distributive vector lattice \(V\) which satisfies two conditions: any interval in \(V\) is sequentially order-compact; every chain in \(V\) is at most countable. This is similar to, but distinct with results of H. H. Schaefer [Math. Ann. 146, 325-330 (1962; Zbl 0102.09905)]: neither contains the other. From this result, the authors derive an integral representation theorem for positive linear operators \(L:C(0,1)\rightarrow V\), which is analogous to the real case.

MSC:
44A60 Moment problems
46A40 Ordered topological linear spaces, vector lattices
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