The purpose of the paper under review is to present an abstract decomposition theorem for a positive finitely additive function on an orthomodular poset taking its values in a Riesz space. More precisely: let \(L\) be an orthomodular poset, let \((V,\|\cdot\|)\) be a Dedekind complete normed Riesz space with order continuous norm and let \(a(L,V)_ +\) be the set of all positive finitely additive functions from \(L\) into \(V\). A non-empty subset \(\mathcal C\) of \(a(L,V)_ +\) is said to be a quasi- cone if \(0\in {\mathcal C}\) and \(\mu_ 1,\mu_ 2\in {\mathcal C}\) imply \(\mu_ 1+ \mu_ 2\in {\mathcal C}\). A quasi-cone \(\mathcal C\) of \(a(L,V)_ +\) is said to be uniformly closed if, for every net \((\mu_ d)_{d\in (D,\geq)}\) in \(\mathcal C\) and every \(\mu\in a(L,V)_ +\) such that \(\|\mu_ d(c)- \mu(c)\|\to 0\) uniformly for \(c\in L\), we have \(\mu\in {\mathcal C}\). If \(\mathcal C\) is a quasi-cone of \(a(L,V)_ +\), an element \(\mu\in a(L,V)_ +\) is said to be \({\mathcal C}\)-singular if \(\nu\in {\mathcal C}\) and \(\mu\leq \mu\) imply \(\nu= 0\).
The main result of the paper can be stated as follows:
Theorem. Let \(\mathcal C\) be a uniformly closed quasi-cone of \(a(L,V)_ +\) and let \(\mu\in a(L,V)_ +\). Then there exists two elements \(\xi\) and \(\eta\) in \(a(L,V)_ +\) such that: i) \(\mu= \xi+ \eta\), ii) \(\xi\in {\mathcal C}\), iii) \(\eta\) is \({\mathcal C}\)-singular.
The paper deduces some consequences of this theorem, namely the Lebesgue and Yosida-Hewitt decomposition theorems given by the first author and the reviewer [Proc. Am. Math. Soc. 120, No. 1, 193-202 (1994; Zbl 0796.28008)].