On recursive decompositions of measures (Q2435818)

Let \((\Omega,\Sigma,\mu)\) be a measure space and let \(\{E_{k}^{m}\}\) \((m\in \mathbb{N})\) be a chain of systems of \(\mu\)-measurable sets with finite positive measure such that \(E_{k}^{m}\cap E_{l}^{m}=\emptyset\) (\(k\neq l\)), but if \(E_{k}^{m}\cap E_{l}^{n}\neq\emptyset\) and \(m<n\), then \(E_{k}^{m}\supset E_{l}^{n}\). Suppose also that \(\nu\) is a complex bounded measure defined on the class \(\Sigma\). In the paper there are given effective conditions providing convergence of the recursive decomposition of the measure \(\nu\) relative to the measure \(\mu\) in terms of the chain of systems \(\{E_{k}^{m}\}\) \((m\in \mathbb{N})\).