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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.

##### MSC:
 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
##### Keywords:
Lebesgue-Bochner space