Integration with respect to a vector measure and function approximation
Let \(\Sigma\) be the \(\sigma\)-algebra of Lebesgue measurable sets of \(\mathbb{R}\), \(\mu\) the Lebesgue measure. Let \((A_i)^n_1\) be a family of pairwise disjoint elements of \(\Sigma\) with \(0<\mu(A_i) <\infty\). The authors define a vector measure as \[ \lambda(A)= \sum^n_1\mu (A_j\cap A)\cdot e_i \] for \(A\in \Sigma\), where \((e_i)^n_1\) is the canonical basis of \(\mathbb{R}^n\). Let \(\Omega= \cup^n_1 A_i\) and \(f\in L^2(\Omega)\); one puts \[ \int_\Omega f d\lambda= \sum^n_1 \Bigl(\int_{A_i} f du\Bigr)e_i \] The authors show that this notion of an integral is useful for approximating elements of \(L^2(\Omega)\) by linear combination of an orthogonal system \((f_i)^n_1\).
scientific article; zbMATH DE number 1780507