Multivariate vector-valued measure (Q2778140)

With author's notations, \(X,Y,U,V,Z\) are Banach spaces and \(L: X \times Y\times U\times V \to Z\) is a mutilinear bounded mapping. \( \Psi , \Omega \) are two sets and \textbf{B , D } algebras of subsets of \( \Psi\) and \( \Omega \) respectively. \textbf{E } denotes the collection of all subsets of \( \Psi \times \Omega \), \textbf{P } = \(\{ E \times F: E \in \mathbf B, F \in \mathbf D \}\), \textbf{Q } is the algebra generated by \textbf{P } and \( \mathbf B \times \mathbf D \) is the \(\sigma\)-algebra generated by \textbf{P }. \( \mu : \mathbf B \to U \) and \( \nu : \mathbf D \to V \) are countably additive measures. \( |\mu |\), \(|\nu |\) denote the usual total variation measures of these measures; if \(|\mu |\) and \(|\nu |\) are finite, \(|\mu |\times |\nu |\), the usual product, can be extended to \( \mathbf B \times \mathbf D \). NEWLINENEWLINENEWLINEThe mapping \( \|\eta \|: \mathbf E \to [0, \infty] \) is defined as: NEWLINENEWLINENEWLINEFor a \( G \in \mathbf Q \), \( \|\eta \|(G)= \sup \|\sum_{i} L(x_{i}, y_{i}, \mu(E_{i}),\nu(F_{i})) \|\) where the sup is taken over all finite collections of disjoint recangles \( E_{i} \times F_{i} \) whose union is contained in \(G\) and \(x_{i}\) and \( y_{i}\) vary in the unit balls of \(X\) and \(Y\); NEWLINENEWLINENEWLINEfor a \( G \notin \mathbf Q \), \( \|\eta \|(G)= \inf \sum_{n=1}^{\infty} \|\eta \|(G_{n})\), inf being taken over all sequences \( \{ (G_{n} \} \subset \mathbf Q \) whose union contains \(G\). NEWLINENEWLINENEWLINEThe mapping \( |\eta |: \mathbf E \to [0, \infty] \) is first defined on \textbf{Q }: \(|\eta |(G)= \sup \sum_{i} \|\mu(E_{i}) \|\|\nu(F_{i})\|\) where the sup is taken over all finite collections of disjoint recangles \( E_{i} \times F_{i} \) whose union is contained in G, and then extended to \textbf{E } in exactly the same way as \( \|\eta \|\). NEWLINENEWLINENEWLINEThe author proves some properties of \( \|\eta \|\) and \( |\eta |\) and measurabe sets and measurable functions which are defined relative to \( \|\eta \|\) and \( |\eta |\). We state the major results of the paper: NEWLINENEWLINENEWLINEI. \( \|\eta \|\) and \(|\eta |\) are outer measures. NEWLINENEWLINENEWLINEII. Suppose \( |\mu |\) and \( |\nu |\) be finite, then \(|\eta |= |\mu |\times |\nu |\) on \( \mathbf B \times \mathbf D \). In this case, when completed, \(|\mu |\times |\nu |\) can be extended to a bigger \(\sigma\)-algebra \(\mathbf Q_{|\mu |\times |\nu |}^{*}\); it is proved that \( |\eta |= |\mu |\times |\nu |\) on \(\mathbf Q_{|\mu |\times |\nu |}^{*}\) ( \( Q_{|\mu |\times |\nu |}^{*}\) is denoted by \(\mathbf Q_{|\eta |}^{*}\)). NEWLINENEWLINENEWLINEIII. Assume \( |\eta |\) and \( \|\eta \|\) to be finite and suppose \( \delta\) stand for \(|\eta |\) or \( \|\eta \|\). Defining the pseudo-metric \(d\) on \textbf{E } as \( d(G_{1},G_{2})= \delta(G_{1} \bigtriangleup G_{2})\), the closure of \textbf{Q }, in \( (\mathbf E , d)\), is denoted by \( \mathbf R_{\delta}\) and its elements are called \(\delta\)-measurable sets. The author proves that \( \mathbf R_{\delta}\) is an algebra and \( \mathbf R_{\delta} \supset \mathbf Q\). If \( \delta = |\eta |\), then \( \mathbf R_{\delta} = \mathbf Q_{|\eta |}^{*}\). Suppose a finite, non-negative measure \(\lambda\) is equivalent to \( \|\eta \|\), then \( \mathbf R_{ \|\eta \|} \supset \mathbf B \times \mathbf D \). These definitions are extended when \( |\eta |\) and \( \|\eta \|\) are not necessarily finite-valued and a few additional results are proved. NEWLINENEWLINENEWLINEIV. Assume \( |\eta |\) and \( \|\eta \|\) be finite-valued. The \(X\)-valued simple functions, relative to \( \delta \), are defined to be of the form \( \sum_{i} x_{i}\chi_{G_{i}} (x_{i} \in X, G_{i} \in \mathbf R_{\delta}) \). An \(X\)-valued function is called \( \delta \)-measurable if there is a sequence of simple functions, converging to it in measure relative to \(\delta\) (converging in measure relative to \(\delta\) has the same meaning as in measure theory). Denoting by \( \eta \{ m \} \) the set of all such measurable functions, among other things, the author proves that \( \|\eta \|\{ m \} \supset |\eta |\{ m \} \). With some readjustments in the definitions, some results are extended when \( |\eta |\) and \( \|\eta \|\) are not necessarily finite-valued.NEWLINENEWLINENEWLINEV. Almost everwhere convergence and almost uniform convergence, relative to \(\delta\) is defined in the usual way. Assuming \( \delta \) be finite-valued, it is easily verified that almost uniform convergence implies almost everwhere convergence and convergence in measure. It is proved that if \( \|\eta \|\) is finite-valued and is equivalent to a finite, non-negative measure, then convergence in measure implies the existence of a subsequence which converges almost everwhere; also convergence almost everywhere implies almost uniform convergence in this case.