UDC 513.881

MATHEMATICS

I. V. Shragin

SPACES GENERATED BY GENFUNCTIONS

(Presented by Academician A. N. Tikhonov on 25 XII 1969)

1°. Let (X) be a space with a (\sigma)-finite complete measure (\mu) ((^{1})), with (0 \leq \mu X \leq \infty); let (S) be the set of all measurable functions on (X) with values in (\overline{R}=[-\infty,\infty]); let (S_f) be the set of almost everywhere (a.e.) finite functions in (S). As usual, functions that coincide a.e. are regarded as equal.

Definition. A function (M(u,x)), (0 \leq u \leq \infty), (x \in X), with values in ([0,\infty]), is called a pregenfunction if it is measurable in (x) for each (u), nondecreasing and left-continuous (in the topology of (\overline{R})) in (u) for almost every (x), and the function (M(0,x)) is summable on (X).

Let (M) be a pregenfunction. Then, if (\varphi \in S), we also have (M(|\varphi(\cdot)|,\cdot)\in S).

Set

[
I_M\varphi=\int_X M[|\varphi(x)|,x]\,d\mu,\qquad
P_M={\varphi\in S_f:I_M\varphi<\infty},\qquad
P_M^\alpha=
]

[
={\varphi\in S_f:\alpha\varphi\in P_M},\qquad
L^M=\bigcup_{\alpha>0}P_M^\alpha,\qquad
L_M^f=\bigcap_{\alpha>0}P_M^\alpha.
]

Since the set (P_M) is convex and symmetric with respect to the zero point (\theta) ((\theta(x)=0) a.e.), (L^M) is a vector space, and (L_M^f) is its subspace. It is obvious that (L_M^f=L^M) if and only if (L^M=P_M), i.e., when (P_M) itself is a vector space.

Set (d_M(x)=\sup{u:M(u,x)<\infty}). It is known ((^{5})) that (d_M\in S). One can show that (L^M={\theta}) if and only if (d_M=0); (L_M^f={\theta}) if and only if (d_M\in S_f).

2°. In their works ((^{10,11})), Musielak and Orlicz introduced on the space (L^M) (which they considered as an example of an abstract modular space) the (F)-norm

[
|\varphi|=\inf{\varepsilon>0:I_M(\varepsilon^{-1}\varphi)\leq \varepsilon},
\tag{1}
]

satisfying the condition: (\lim|\varphi_n|=0) if and only if (\lim I_M(\alpha\varphi_n)=0) for every (\alpha>0) (any (F)-norm in (L^M) possessing this property will be called normal). In ((^{10,11})) it is assumed that the pregenfunction (M) takes finite values for all ((u,x)\in[0,\infty)\times X), is continuous in (u) for each (x), and (M(u,x)=0) if and only if (u=0). It turns out that these conditions can be weakened. To this end we give the following

Definition. A pregenfunction (M) is called a genfunction if (M(0,x)=0) and (M(\infty,x)>0) a.e. on (X), while (M(+0,x)=0) a.e. on ({x:d_M(x)>0}).

Theorem 1. The space (L^M) admits the introduction of a normal (F)-norm if and only if (M) is a genfunction. In particular, if (M) is a genfunction, then formula (1) defines in (L^M) a normal (F)-norm, called the Musielak–Orlicz (F)-norm.

Let us note that (|\varphi|) in (1) is meaningful for every (\varphi\in S), but if (\varphi\in S\setminus L^M), then (|\varphi|) may be infinite. It is not difficult to see that the (F)-norm (1) has the property of monotonicity (if a.e. (|\varphi(x)|\leq|\psi(x)|), then (|\varphi|\leq|\psi|)) and of left monotone continuity (if a.e. (|\varphi_n(x)|\uparrow|\varphi(x)|), then (|\varphi_n|\uparrow|\varphi|)). Further, for any normal (F)-norm in (L^M), convergence in the (F)-norm implies convergence in measure on every subset of finite measure (but not necessarily in measure on all of (X), as is the case when the genfunction (M) is a function only of (u)). With the aid of this property one proves

Theorem 2. Let (M) be a genfunction. Then the space (L^{M}), with respect to every normal (F)-norm, is complete, i.e., is an (F)-space, and any two normal (F)-norms in (L^{M}) are topologically equivalent.

Theorem 3. If (M) is a genfunction, then, for any normal (F)-norm, the subspace (L_{M}^{f}) is closed and coincides with the set of elements of the space (L^{M}) having absolutely continuous (F)-norms.

We give some examples of spaces (L^{M}).

1. If (M(u,x)=u^{p}\rho(x)), where (0<p<\infty), (\rho(x)) is a positive measurable function, then (L^{M}) is the space (L^{p}) with weight (\rho(x)).

2. Let (M(u,x)=\varphi(u)), where the function (\varphi) is continuous, nondecreasing, vanishes only at zero, and (\varphi(u)\to\infty) as (u\to\infty). Then (L^{M}) is the generalized Orlicz space ((^{9})).

3. If a.e. (0<d_{M}(x)<\infty) and (M(u,x)=0) for (0\le u\le d_{M}(x)), then
   [
   L^{M}={\varphi\in S_{f}:\operatorname{vrai\,sup}|\varphi(x)|\cdot(d_{M}(x))^{-1}<\infty}.
   ]
   Moreover, (|\varphi_{n}|\to0) if and only if
   [
   |\varphi_{n}(x)|\cdot(d_{M}x)^{-1}\to0
   ]
   uniformly a.e.

4. If (M) is a pregenfunction for which the function (M(\infty,x)) is summable on (X), then (L^{M}=S_{f}); if, in addition, (M) is a genfunction, then convergence in (L^{M}) with respect to the normal (F)-norm is equivalent to convergence in measure on every subset of finite measure.

3°. If the pregenfunction (M) is a function only of (u) and (d_{M}>0) ((d_{M}=\infty)), then every measurable function bounded on (X), distinct from zero on a set of finite measure, is contained in (L^{M}) (respectively in (L_{M}^{f})). If, however, (M) is a function of (u) and (x), then this fact, generally speaking, does not hold. For example, if (X=(0,1)), (\mu) is Lebesgue measure, (M(u,x)=ux^{-1}), then nonzero constants do not belong to (L^{M}=P_{M}). However, every measurable bounded function that vanishes in a neighborhood of zero belongs to this space. It turns out that, also in the general case, as follows from what follows, there exist classes of bounded functions contained in (L^{M}) and (L_{M}^{f}).

We shall call a chain ((\pi)) a nondecreasing sequence of measurable sets (\pi_{n}\subset X) for which (\mu\pi_{n}<\infty), (n=1,2,\ldots).

Lemma 1 (cf. ((^{8}))). Let (\mathfrak A) be a nonempty family of measurable subsets of a measurable set (Y\subset X), and suppose the following conditions are fulfilled:
1) if (E_{1},E_{2}\in\mathfrak A), then (E_{1}\cup E_{2}\in\mathfrak A);
2) if (E\subset Y) and (\mu E>0), then there exists an (F\subset E) such that (F\in\mathfrak A) and (\mu F>0).
Then there exists a chain ((\pi)) such that all (\pi_{n}\in\mathfrak A) and
[
\mu\bigl(Y\setminus \lim \pi_{n}\bigr)=0.
]

For a given chain ((\pi)), we shall call a function (\varphi) measurable on (X) ((\pi))-bounded if
[
\operatorname{vrai\,sup}|\varphi(x)|<\infty
]
and (\varphi(x)=0) a.e. on (X\setminus\pi_{n}), beginning with some (n). With the help of Lemma 1 one proves

Theorem 4. If (M) is a pregenfunction, then there exists a chain ((\pi)) such that
[
\lim \pi_{n}={x:d_{M}(x)>0}
\quad
\bigl(\lim \pi_{n}={x:d_{M}(x)=\infty}\bigr),
]
and all ((\pi))-bounded functions are contained in (L^{M}) (respectively in (L_{M}^{f})).

Let now (M) be a genfunction, and let (L_{M}^{\pi}) be the closure of the set of all ((\pi))-bounded functions contained in (L^{M}), with respect to any normal (F)-norm. Obviously, (L_{M}^{\pi}) is a subspace of (L^{M}). It is not difficult to show that if
[
\mu\bigl({x:d_{M}(x)=\infty}\setminus \lim \pi_{n}\bigr)=0,
]
then (L_{M}^{f}\subset L_{M}^{\pi}). Moreover, as follows from Theorem 4, there always exists a chain ((\pi)) such that
[
\lim \pi_{n}={x:d_{M}(x)=\infty}
]
and (L_{M}^{f}=L_{M}^{\pi}). Relying on this assertion, one can show that if the measure (\mu) has a countable basis ((^{1})), then (L_{M}^{f}) is separable.

4°. Here we shall consider some metric properties of the space (L^{M}) endowed with the (F)-norm (1). First of all, note that if (\varphi\in P_{M}), then
[
|\varphi|\le |d_{M}|,
]
as follows from the monotonicity of the (F)-norm (1).

Lemma 2. If (M) is a pregenfunction, then there exists a nondecreasing sequence of nonnegative functions (\varphi_{n}\in P_{M}) such that
[
\lim \varphi_{n}(x)=d_{M}(x)\quad \text{a.e.}
]

Theorem 5. If (M) is a genfunction, then
[
|d_{M}|=\sup{|\varphi|:\varphi\in P_{M}}.
]

Remark 1. From Lemma 2 there also follows the following proposition, more general than Theorem 5. Let (f) be a functional on (S) with values

on ([0,\infty]), possessing the property of monotone left-continuity. Then, if (M) is a pregenfunction, then

[
f(d_M)=\sup {f(\varphi):\varphi\in P_M}.
]

A natural supplement to Theorem 5 is

Theorem 6. Let (M) be a genfunction. Then
[
{\varphi\in L^M:|\varphi|\leq 1}\subset
{\varphi\in L^M:I_M\varphi\leq 1}\subset P_M;
]
moreover,
[
P_M={\varphi\in L^M:|\varphi|\leq 1}
]
if and only if (I_M(d_M)\leq 1).

Let us now consider the question of the position of the set (P_M) relative to the subspace (L_M^f). To this end put
[
\rho(\varphi,L_M^f)=\inf {|\varphi-\psi|:\psi\in L_M^f}.
]

Theorem 7. Let (M) be a genfunction. Then
[
{\varphi\in L^M:\rho(\varphi,L_M^f)<1}\subset P_M.
]
Moreover, if (\mu{x:0<d_M(x)<\infty}=0), then
[
P_M\subset {\varphi\in L^M:\rho(\varphi,L_M^f)\leq 1}.
]

5°. Let the genfunction (M) be convex in (u) for almost every (x). Then (M(\infty,x)=\infty) a.e. and the function (M) is continuous in (u) on ([0,d_M(x))) for almost every (x) for which (d_M(x)>0). Such a genfunction is called a Young function, and the space (L^M) generated by the Young function will be called (5) an Orlicz–Nakano space, since this space was first described in (12), and a particular case of it (when (M) is a function only of (u)) is an Orlicz space, more precisely an Orlicz space in the sense of Zaanen (14, 15) (cf. (2, 3)).

The Orlicz–Nakano space is a Banach space with norm
[
|\varphi|_1=\inf {\varepsilon>0:I_M(\varepsilon^{-1}\varphi)\leq 1}
]
or
[
|\varphi|_2=\inf {\alpha^{-1}(1+I_M(\alpha\varphi)):0<\alpha<\infty}
]
(for (|\varphi|_2) there is also another expression (2, 5), using the complementary Young function). Both norms are normal and, consequently, topologically equivalent to the (F)-norm (1) (see also (10), where inequalities are established for the (F)-norm (1) and the norm (|\cdot|_1)). We also note that, as in Orlicz spaces,
[
|\varphi|_1\leq |\varphi|_2\leq 2|\varphi|_1.
]

The basic facts of the theory of Orlicz spaces extend to Orlicz–Nakano spaces. Here we shall give several propositions generalizing some results from (4, 6).

Let (M) be a Young function;
[
\rho_k(\varphi,L_M^f)=\inf {|\varphi-\psi|_k:\psi\in L_M^f},
]
[
\Pi_k={\varphi\in L^M:\rho_k(\varphi,L_M^f)<1},\quad k=1,2.
]
It is easy to see that
[
\overline{\Pi}_k={\varphi\in L^M:\rho_k(\varphi,L_M^f)\leq 1},\quad k=1,2.
]

Theorem 8. (\Pi_2\subset \Pi_1\subset P_M).

Put
[
X^0={x:0<d_M(x)<\infty}.
]

Theorem 9. If (\mu X^0=0), then
[
\Pi_1=\Pi_2=\operatorname{int} P_M,\qquad
\overline{\Pi}_1=\overline{\Pi}_2=\overline{P}_M.
]

Corollary. If (\mu X^0=0), then
[
\rho_1(\varphi,L_M^f)=\rho_2(\varphi,L_M^f)=\inf {\alpha>0:\alpha^{-1}\varphi\in P_M}
]
for every (\varphi\in L^M), i.e. the norms (|\cdot|_1) and (|\cdot|_2) generate one and the same norm in the quotient space (L^M/L_M^f) (cf. (7), p. 8).

Theorem 10. If (\mu X^0>0), then
[
\max {r:D_r\subset P_M}=1,
]
where
[
D_r={\varphi:|\varphi|_1\leq r}.
]
Moreover, if (I_M(d_M)\leq 1), then (|d_M|_1=1) and (P_M=D_1); if (I_M(d_M)>1), then (|d_M|_1>1) and both inclusions
[
D_1\subset P_M\subset {\varphi:|\varphi|_1\leq |d_M|_1}
]
are proper.

Remark 2. Since the norms (|\cdot|_1) and (|\cdot|_2) have the property of monotone left-continuity, by Remark 1,
[
|d_M|_k=\sup {|\varphi|_k:\varphi\in P_M},\quad k=1,2.
]
It follows that the set (P_M) is bounded in the Orlicz–Nakano space (L_M) if and only if (d_M\in L^M).

In conclusion we note that a number of properties of the spaces (L_M) (conditions for the embedding of one space in another, criteria for closedness and openness of the set (P_M), etc.) can be obtained in the form of simple consequences of certain propositions on the Nemytskii operator, whose investigation in spaces generated by genfunctions is carried out in (13).

I sincerely thank M. M. Vainberg for valuable advice concerning this work.

Proof correction note. After the manuscript had been submitted for publication, the author became acquainted with the work (^{(16)}), in which spaces generated by genfunctions are considered under the condition that (d_M(x)>0) for all (x).

Tambov Institute
of Chemical Machine-Building

Received
16 XI 1969

REFERENCES

(^{1}) A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Moscow, 1968.
(^{2}) V. R. Portnov, DAN, 170, No. 6 (1966).
(^{3}) V. R. Portnov, DAN, 175, No. 2 (1967).
(^{4}) I. V. Shragin, Scientific Notes of Kishinev Univ., 91, 81 (1967).
(^{5}) I. V. Shragin, ibid., 91, 91 (1967).
(^{6}) I. V. Shragin, DAN, 179, No. 5 (1968).
(^{7}) T. Andô, Nieuw Arch. Wiskunde, 8, 1 (1960).
(^{8}) W. A. J. Luxemburg, A. C. Zaanen, Proc. Nederl. Akad. Wet., A59, No. 1, 110 (1956).
(^{9}) W. Matuszewska, Bul. Acad. Pol. sci., Sér. sci. math., astr., phys., 8, No. 6, 349 (1960).
(^{10}) J. Musielak, W. Orlicz, ibid., 7, No. 11, 661 (1959).
(^{11}) J. Musielak, W. Orlicz, Stud. math., 18, No. 1, 49 (1959).
(^{12}) H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo, 1950.
(^{13}) I. V. Shragin, DAN, 189, No. 1 (1969)*.
(^{14}) A. C. Zaanen, Linear Analysis, N. Y., 1953.
(^{15}) A. C. Zaanen, Proc. Nederl. Akad. Wet., A52, No. 5–6, 488 (1949).
(^{16}) T. Itô, J. Fac. Sci. Hokkaido Univ., ser. I, Math., 15, No. 3, 4 (1961).

* Correction. In article (^{(13)}) the following corrections must be made:

p. 64, formula (1) should read

[
\Phi[\beta|g(u,x)|,x]\leq \gamma M[\alpha|u|,x]+f(x).
\tag{1}
]

p. 64, line 25 from the bottom: printed (h[P_M^\alpha(\Delta)L^\Phi), should read (h[P_M^\alpha(\Delta)]\subset L^\Phi).

p. 64, line 21 from the bottom: printed “то,” should read “что.”

p. 66, line 4, should read

[
\Phi[\beta|g(u'',x)-g(u',x)|]\leq \gamma[M(|u'|,x)+M(\alpha|u''-u'|,x)]+f(x).
]

p. 66, line 5: printed (\alpha), should read (a).