Bilinear integration in tensor products
Let \(X\), \(Y\) be Banach spaces with \(X\otimes Y\) their tensor product. If \(\tau\) is a norm on \(X\otimes Y\), denote the space \(X\otimes Y\) with this norm by \(X\otimes_\tau Y\) and its completion by \(X\widehat\otimes_\tau Y\). Call \(\tau\) a norm tensor product if there exists \(C>0\) such that (i) \(\tau(x\otimes y)\leq C\|x\|\|y\|\) for all \(x\in X\), \(y\in Y\) and (ii) \(X'\otimes Y'\) can be identified with a linear subspace of \((X\otimes_\tau y)'\) with \(\|x'\otimes y'\|\leq C\|x'\|\|y'\|\) for all \(x'\in X'\), \(y'\in Y'\). If \(\tau\) is a norm tensor product, then \(\tau\) is completely separated if \(X'\times Y'\) separates \(X\widehat\otimes_\tau Y\); for example, if \(\tau\) is the injective tensor product, this always holds, and if \(\tau\) is the projective tensor product, this holds if \(X\) or \(Y\) has the approximation property. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(S\) and \(m:\Sigma\to Y\) be countably additive. If \(\varphi= \sum^n_{j=1} x_j\chi_{A_j}: S\to X\) is an \(X\)-valued \(\Sigma\)-simple function, the integral of \(\varphi\) with respect to \(m\) over \(A\in\Sigma\) is defined by \[ \int_A \varphi\otimes dm= \sum^n_{j= 1}x_j\otimes m(A\cap A_j). \] Under the assumption that \(\tau\) is completely separating, a function \(f:S\to X\) is \(m\)-integrable if there exist \(X\)-valued \(\Sigma\)-simple functions \(\{\varphi_k\}\) such that \(\varphi_k\to f\) \(m\)-a.e. and \(\left\{\int_A \varphi_k\otimes dm\right\}\) converges in \(X\widehat\otimes_\tau Y\) for every \(A\in\Sigma\). The integral of \(f\) with respect to \(m\) over \(A\in \Sigma\) is defined to be \(\int_A f\otimes dm= \lim\int_A \varphi_k\otimes dm\). The authors show that the integral is well-defined and then very thoroughly investigate its important properties and compare it with other integrals. For example, if \(m\) is scalar-valued, the integral is equivalent to the Pettis integral for strongly measurable functions. More generally, the authors give a very careful comparison of their integral to the integrals of Bartle and Dobrakov which treat the integration of vector-valued functions with respect to operator-valued measures. They show that their integral is more natural and more general in this setting. There are numerous examples given which illustrate the technical differences between their integral and those of Bartle and Dobrakov.
Pettis integral for strongly measurable functions
integration of vector-valued functions with respect to operator-valued measures
scientific article; zbMATH DE number 1309729