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Some equalities among Orlicz spaces. I. (English) Zbl 0742.46014

A function \(\varphi: [0,\infty)\to[0,\infty]\) is called an Orlicz function if it is vanishing and continuous at zero, nondecreasing and not identically equal to zero. An Orlicz function \(\varphi(u)\) continuous for all \(u\geq 0\), taking only finite values, vanishing only at zero and tending to \(\infty\) as \(u\to\infty\) is called a \(\varphi\)-function.
Let \(L^ \varphi\) and \(E^ \varphi\) denote the Orlicz space and the space of all finite elements associated with an Orlicz function \(\varphi\) respectively, on an arbitrary measure space. Let \(\Phi_ 1\) be the set of all Orlicz functions taking only finite values and such that \(\varphi(u)\to\infty\) as \(u\to\infty\) and let \(\Phi_ 2\) be the set of all Orlicz functions vanishing only at zero and such that \(\varphi(u)\to\infty\) as \(u\to\infty\).
The main results of the paper are the following: for each \(\varphi_ 1\in\Phi_ 1\) and \(\varphi_ 2\in\Phi_ 2\) there exist sets of \(\varphi\)-functions \(\Psi_ 1^{\varphi_ 1}\) and \(\Psi_ 2^{\varphi_ 2}\) respectively, such that the following equalities hold: \[ E^{\varphi_ 1}=\bigcup_{\psi\in\Psi_ 1^{\varphi_ 1}} E^ \psi=\bigcup_{\psi\in\Psi_ 1^{\varphi_ 1}}L^ \psi \] and \[ L^{\varphi_ 2}=\bigcap_{\psi\in\Psi_ 2^{\varphi_ 2}} L^ \psi=\bigcap_{\psi\in\Psi_ 2^{\varphi_ 2}}E^ \psi. \] Moreover, if the measure spaces is nonatomic or purely atomic, then the strict inclusions hold: \(L^ \psi\subset E^{\varphi_ 1}\) for each \(\psi\in\Psi_ 1^{\varphi_ 1}\) and \(L^{\varphi_ 2}\subset E^ \psi\) for each \(\psi\in\Psi_ 2^{\varphi_ 2}\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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