Representation of additive functionals on non-solid Köthe spaces (Q1174464)

In the results of this paper, the author has considered conditions under which norm convergence implies convergence in measure in some general vector function spaces which are defined as Köthe spaces. If \((T,\Sigma,\mu)\) is a \(\sigma\)-finite measure space and \((X,\|\cdot\|)\) is a Banach space, let \(M(T,X)\) denote \(\{f: T\rightarrow X: f\hbox{ is a.e. finite and strongly measurable}\}\). The normed space \((L,\|\cdot\|_ L)\) is said to be a normed Köthe space if {(i) } \(\chi_ Af\in L\) for \(A\in\Sigma\), \(f\in L\), {(ii) } \(A,B\in\Sigma\), \(A\subseteq B\) implies that \(\|\chi_ A f\|_ L\leq \|\chi_ B f\|_ L\). It is shown in the main theorem of the paper that if \(\mu(T)<\infty\), \(L\) satisfies a condition (+): (for \(g\in L\), with \(\mu(\hbox{supp }g)<\infty\), sequence \(\{f_ n\}\), with \(\| f_ n\|_ L\rightarrow 0\) (as \(n\rightarrow\infty\)), and sets \(A_ n=\{t\in\hbox{supp }g:\| f_ n(t)\|\geq\| g(t)\|\}\), there is a Köthe overspace \(\Lambda\) such that \(\|\chi_{A_ n}g\|_ \Lambda\rightarrow 0\) (as \(n\rightarrow\infty\))), then norm convergence in \(L\) implies convergence in measure. The main result is applied to derive representation theorems for a closed decomposable subset of \(E\times L_ 1\), where \(E\) satisfies condition (+) and \(L_ 1\) consists of integrable functions from \(T\) into the real line.