zbMATH — the first resource for mathematics

Some properties of Bochner integral in bitopological vector spaces and introduction to generalized Lebesgue spaces \(L^p(E,(X_{\theta},\| .\|))\). (English) Zbl 1120.28004
Let \((X,\theta)\) be a separated topological vector space (with a given separating family of quasi-norms) and let \(\| .\| \) be the norm defined on \(X\). Let \(\sigma(\theta)\) be any locally convex topology on \(X\) generated by a family of continuous semi-norms on \((X,\theta)\). Let us assume that the bitopological space \((X,\theta,\sigma(\theta))\) is sequentially complete, i.e., every Cauchy sequence in \((X,\theta)\) is convergent in \((X,\sigma(\theta))\), the unit ball of \((X,\| .\| )\) is closed in \((X,\sigma(\theta))\), and every bounded set in \((X,\| .\| )\) is bounded in \((X,\theta)\). Under these hypotheses the authors present some properties of the Bochner integral with respect to the pair of topologies \(\theta\) and \(\| .\| \) and they introduce a class of integrable functions \(L^p(E,(X_\theta,\| .\| ))\) where \(X_\theta\) denotes the space \((X,\theta)\) which includes the Lebesgue-Bochner space \(L^p(E,(X,\| .\| ))\). Let us recall that \(L^p(E,(X_\theta,\| .\| ))\) is the class of measurable functions \(\varphi:E\to X_\theta\) where “\(\varphi\) is measurable” means that for every \(\varepsilon>0\) there exists a compact set \(K\subset E\) such that \(\mu(E\setminus K)<\varepsilon\) and \(\varphi| _K\) is continuous. Here \(\mu\) means the Lebesgue measure. The authors give an example showing that in general the inclusion \(L^p(E,(X,\| .\| ))\subseteq L^p(E,(X_\theta,\| .\| ))\) is strict.

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A25 Integration with respect to measures and other set functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions