Unions and Intersections of Families of Lp-Spaces

Let L^Ψ and E^Ψ be the ORLICZ space and the space of finite elements respectively, on a measure space (Ω, Σ, μ), and let T ? (0, ∞). It is proved that if inf {p: p ? T} ? T or sup {p: p ? T} ? T and μ is an infinite atomless measure, then there is no ORLICZ function Ψ such that: documentclass{article}pagestyle{empty}begin{document}$ L^varphi = Linmathop { cup L^p }limits_{pvarepsilon T} $end{document} or documentclass{article}pagestyle{empty}begin{document}$ E^varphi = Linmathop { cup L^p }limits_{pvarepsilon T} $end{document} and moreover, there is no ORLICZ function Ψ such that: documentclass{article}pagestyle{empty}begin{document}$ L^varphi = Linmathop { cap L^p }limits_{pvarepsilon T} $end{document} or documentclass{article}pagestyle{empty}begin{document}$ E^varphi = Linmathop { cap L^p }limits_{pvarepsilon T} $end{document}.