## 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:

 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

Orlicz function; Orlicz space