Dependence on p and Ω, Local Spaces

This chapter shows that Lebesgue space L p (Ω;E) increases as p decreases provided that the measure of Ω is finite. It also states the Riesz convexity theorem on the dependence in p for an arbitrary domain, and defines local Lebesgue spaces and their properties. The chapter also defines the localized-extension of a measure. It explains that p -integrable measures possess the finite gluing property. The chapter further presents some additivity properties of the semi-norms relative to the domain. It describes an infinite-gluing property for an increasing sequence of open sets.