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Владикавказский математический журнал 2017, Том 19, Выпуск 3, С. 41-50
УДК 517.98
MAXIMAL QUASI-NORMED EXTENSION OF QUASI-NORMED LATTICES
A. G. Kusraev and В. B. Tasoev
To Professor A. B. Shabat on occasion of his 80th birthday
The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension XK of a Dedekind complete quasi-normed lattice X with the weak a-Fatou property is a quasi-Banach lattice if and only if X is intervally complete. Moreover, XK has the Fatou and the Levi property provided X
this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.
Mathematics Subject Classification (2010): 46A16, 46B42, 46E30, 46G10, 47B38, 47G10.
Key words: quasi-Banach lattice, maximal quasi-normed extension, Fatou property, Levi property vector measure, space of weakly integrable functions.
1. Introduction
For over recent 25 years the spaces of integrable functions with respect to a measure taking values in a Banach (quasi-Banach) lattice have been a field of increased interest. The spaces of integrable and weakly integrable functions with respect to a vector measure possess interesting order and metric properties and have been studied intensively by many authors. They find applications in important problems such as the representation of abstract quasi-Banach lattices as spaces of integrable functions, the study of the optimal domain of linear operators, domination and factorization of operators, spectral integration etc., see [4, 5, 7, 18, 20] and the references therein.
A key role in the theory is played by the space L^ (ц) of weakly integrable functions with respect to a measure ц with values in a Banach space, see the survey paper by Curbera and Ricker [5] and the book by Okada, Ricker and Sánches Pérez [18]. However, in the context of quasi-Banach spaces, when the conjugate space may turn out to be trivial, the duality based definition of Lw (ц) does not work, so we need to find a suitable substitute for Lw (ц)-
In the case of a measure taking values from et CjlletSl- Banach lattice two natural candidates for the space of weakly integrable function were indicated in [10]. The first one arises as the domain of the smallest extension of the integration operator (see Aliprantis and Burkinshaw [3, Theorem 1.30]), and the second one is based on the construction of the maximal normed extension introduced by Abramovich in [1]. The approach based on the smallest extension of the integration operator is presented in [11].
© 2017 Kusraev A. G. and Tasoev В. B.
In order to realize the second possibility, it is necessary to extend Abramovich's construction to quasi-Banach setting, as done in this article. In Section 2 we sketch the needed information concerning quasi-Banach lattices and prove some Riesz-Fischer type completeness theorems for quasi-normed lattices; next, we gave a characterization of order continuous quasi-Banach lattices. In Section 3 we examine the construction of the maximal quasi-normed extension introduced by Abramovich [1] for Banach lattices. It is proved that the maximal quasi-normed extension XK of a Dedekind complete quasi-normed lattice X with the weak a-Fatou property IS el CjlletSl- X
XK has the Fatou and the Levi property provided that X is a Dedekind complete quasi-normed space with the Fatou property.
We use the standard notation and terminology of Aliprantis and Burkinshaw [3] and Meyer-Nieberg [17] for the theory of vector and Banach lattices (see also Abramovich and Aliprantis [2], Luxemburg and Zaanen [13]). Throughout the text we assume that all vector spaces are defined over the field of reals and all vector lattices are Archimedean. We let := denote the assignment by definition, while N and R symbolize the naturals and the reals.
2. Quasi-Banach Lattices
In this section, we briefly sketch the needed information concerning quasi-Banach lattices. In particular, we give some simple results on the completeness and order continuity of quasi-Banach lattices for which we have not found references.
Definition 2.1. A quasi-normed space is a pair (X, || ■ ||) where X is a real vector space and || ■ || is a quasi-norm, a function from X to R such that the following conditions hold:
(1) ||x|| ^ 0 te all x G X and ||x|| = 0 if and only if x = 0.
(2) ||Ax|| = |A|||x|| for all x G X Mid A G R.
(3) There exists a constant C ^ 1 with ||x + y|| ^ C(||x|| + ||y||) for all x, y G X. If, in addition, for some 0 < p ^ 1 the inequality
(4) ||x + y||p < ||x||p + ||y||p holds for all x, y G X,
then || ■ || is called a p-norm and (X, || ■ ||) is called a p-normed space.
C
multiplier, or modulus of concavity of the quasi norm. Note that || n=i xk|| ^ n=i Ck||xk || for all x1;..., xn G X.
Two quasi-norms || ■ || and || ■ ||' are equivalent if there is a constant A ^ 1 such that A-1 ||x|| ^ ||x||' ^ A||x|| to all x G X. By the Aoki-Rolewicz theorem (see [8]), each quasinorm is equivalent to some p-norm for so me 0 < p ^ 1.
Theorem 2.2 (Aoki-Rolewicz). Let (X, || ■ ||) be a quasi-normed space with the quasi-triangle constant C ^ 1 and p = (1 + log2 C)-1. Define || ■ ||p : X ^ R as
HP:= inf : x ^^xk, n G (x G X).
k=1
Then 0 < p ^ 1, || ■ ||p is a^^^^™, and ||xyp ^ ||x|| ^ for all x G X.
< See Maligranda ^^^^rem 1.2], Pietsch [19, 6.2.5]. >
Thus, we may assume unless otherwise is mentioned that et QUetSl- Banach space is equipped with a p-norm for so me 0 < p ^ 1.
x
A topological vector space X is said to be locally bounded if it has a bounded neighborhood of zero. A quasi-normed space is a locally bounded topological vector space if we take the sets {x G X : ||x|| ^ e} (0 < e G R) for a base of neighborhoods of zero. Moreover, this topology may be induced by metric d(x, y) := | | | x — y| | | p (x, y G X) where | | | ■ | | | is an equivalent p-norm.
X
can be deduced from a quasi-norm, which may be obtained as the Minkowski functional of a bounded balanced neighborhood B of zero:
||x|| := ||x||b := inf {0 < A G R : x G AB} (x G X).
A quasi-norm may be discontinuous in its own topology [19, 6.1.9]. However, every quasi-norm is equivalent to a continuous one, since a p-norm is continuous.
Definition 2.3. A quasi-Banach space (p-normed space) IS el CjlletSl- normed space which is complete in its metric uniformity.
Theorem 2.4. A quasi-normed space X := (X, || ■ ||) with a triangle constant C ^ 1 is complete (and hence a quasi-Banach space) if and only if for every series (xk) in X such that J2kLi Ck||xk|| < to there exists J2kLi xk G X and
<x <x
exk <£ Ck+l
k=l
k=l
< See Maligranda [15, Theorem 1.1]. >
The basic results of the Banach space theory such as open mapping theorem and the closed graph theorem (for linear operators) are valid also in the context of quasi-Banach spaces, see [9].
Definition 2.5. A quasi-Banach (quasi-normed, p-Banach) space (X, || ■ ||) is called a quasi-Banach lattice (respectively, quasi-normed lattice, p-Banach lattice) if, in addition, X is a vector lattice and |x| ^ |y| implies ||x|| ^ ||y|| for all x,y g X.
Lemma 2.6. In any quasi-normed lattice X lattice operations are continuous and the positive cone is closed. Moreover, if an increasing (decreasing) net (xa)aeA is quasi-norm convergent to x g X, then x = supaeA xa (x = infxa).
< This can be ensured just as in the case of Banach lattice using monotonicity of the quasi-norm and quasi-triangle inequality. >
X IS et QUetSl-
X
that Amemiya's result on completeness of normed lattices is true in the context of quasi-
X
X
result.
Theorem 2.7. For a quasi-normed space X := (X, || ■ ||) with a triangle constant C ^ 1 the following assertions are equivalent:
(1) X
(2) For every series (xk) in X+ such that J^kLi Ck|xk|| < to there exists x G X with k= 1 xk-
(3) For every series (xk) in X+ such that J2k=1 Ck|xk|| < to there exists x G X with
E^ _
k= 1 xk := SUpraeN k= 1 xk-
< See [12, Theorem 2.7]. >
Definition 2.8. A quasi-Banach lattice (X, || ■ ||) (as well as the quasi-norm || ■ ||) is said to be order continuous, if xa I 0 implies ||xa|| I 0 for any net (xa)aeA in X. If arbitrary nets are replaced by sequences, one speak of order a-continuity.
X
X
(2) Every increasing order bounded sequence in X+ is convergent.
(3) X is Dedekind a-complete and order a-continuous.
< See [12, Theorem 2.10]. >
Definition 2.10. A quasi-Banach lattice (X, || ■ ||) is said to have the weak Fatou property (respectively weak a-Fatou property) if there exists K > 0 (called the weak Fatou constant) such that for every increasing net (xa) (respectively sequence (xn)) with the supremum x g X we have ||x|| ^ Ksupa ||xa|| (respectively ||x|| ^ Ksupn ||xn||). If K = 1 then ||x|| = supa ||xa|| Xa
Definition 2.11. Say that et CjlletSl- normed lattice (X, || ■ ||) has the Levi property (respectively a-Levi property) if supa xa (respectively supn xn) exists for every increasing net (xa) (respectively sequence (xn)) in X+ provided that supa ||xa|| < to (respectively supn ||xn|| < to). A gwasi-KB-space is an order continuous quasi-normed lattice with the Levi property.
X
X
< The fact that a quasi-normed lattice with the Levi property has also the weak Fatou property is the only thing that needs verification. The proof is similar to that of Proposition 2.4.19 in Meyer-Nieberg [17].
X
n G N there exists an increasing net (yn,a)aeA(n) ™ X+ such that = supaeA(n) yn,a exists and
||yn| ^ nr, T = Cnn2supaeA(„) HVnaH (n G N),
where C ^ 1 is the triangle const ant of X. Putting := yn/T, yn,a := yn)Q:/T we arrive at the following relations:
j/n =SUpaeA(„) ||j/n|| ^ n, ||j/ra>Q|| < C-nn-2 (n G N).
Let (xY) stands for the net of finite suprema of elements in : n G N, a G A(n)}.
If x7 = yni,ai V ■ ■ ■ V h aj G A(nj), then
k k X 1
||®71| < \\ynim +■■■ + Vnk,ak\ | < ^2cj\\ynjtaj II < " "/"'
j=1 j=1 n=1
By hypothesis, x = supY xY exists and satisfies x ^ for all n G N. Consequently, ||x|| ^ n for all n G N a contradiction. >
3. Maximal Quasi-Normed Extension
Consider et QUetSl- normed lattice (X, || ■ ||) with the quasi-triangle constant C. Let Xd
XX
sublattice of X5, while X5 itself is a Dedekind complete vector lattice. Define a function || ■ ||5 : X5 ^ R as
||x||5:= inf {||x|| : x G X+, |x| ^ x} (x G X5).
Clearly, ||x|| = ||x||5 for all x G X and ||x||5 < ro for each x G X5, since X is majorizing sublattice. Positive homogeneity and monotonicity of || ■ ||5 are obvious. Moreover, if |x| ^ x and |y| ^ y for some x, y G X and x, y G X5, then |x + y| ^ x + y and ||x + y||5 ^ ||x + y|| ^ C(|x|| + ||y||) and hence ||x+y||5 ^ C(||x||5 + ||y||5)■ It follows that (X5, ||-||5) IS et QUetSl- normed lattice with the same quasi-triangle constant.
Lemma 3.1. If (X, || ■ ||) is a quasi-Banach lattice with a triangle constant C or a p-Banach lattice, then so is (X5, || ■ ||5).
< Assume that YlkLi Ck||xk||5 < ro for a sequence (xk) in X+. Pick xk G X+ such that xfc < xk and ||xk|| < ||xk||5 + 1/(2C)k. Then
n n n 1
k=1 k=1 k=1
and hence J2'k=1 Ck||xk|| < ro. By Theorem 2.6 x := 0-^xk exists in X. Consequently, xk exists in X5, since Y^k=1 xk ^ x for all n G N > Assume now that (X, || ■ ||) is a Dedekind complete quasi-normed lattice with
a quasi™
triangle constant C. Identify X with an order dense ideal in its universal completion Xu. Define a function || ■ ||K : Xu ^ R U {+ro} by putting
||x|K:= sup {||x|| : x G X, 0 ^ x ^ |x|} (x G Xu).
Observe that ||x|| = ||x||K for all x G X. Denote XK := {x G Xu : ||
x||K < ro}. If 0 ^ u ^
|x + y| ^ |x| + |y| for some x, y G X^d u G X, then there exist x, y G X with 0 ^ x ^ |x|, 0 ^ y ^ |y|, and u = x + y. It follows that ||u|| ^ C(||x|| + ||y||) ^ C(||x||K + 11y|K) and thus ||x + y||K ^ C(|x||K + ||y||K)■ Similarly, || ■ ||K is a p-norm, whenever || ■ || is. Taking into account obvious monotonicity and positive homogeneity of || ■ ||K, we see that (XK, || ■ ||K) is a quasi-normed lattice with the quasi-triangle constant C and, if || ■ || is a p-norm, so is || ■ ||K. Definition 3.2. A maximal quasi-normed extension of
et QUetSl- normed lattice (X, || ■ ||)
is the pair (X5k, || ■ ||5*) with X5k := (X5)K and
||x||5* := supj inf{||x|| : x G X, |x| < x} : x G X5, 0 < x < |x|} (x G X5k).
Observe that if X is Dedekind complete then X5k = XK and || ■ ||5K = || ■ ||K.
Lemma 3.3. If (X, || ■ ||X) and (Y, || ■ ||y) are quasi-normed lattices, Y is an order dense ideal in Xu containing X, and ||x||X = ||x||y for all x G X, then Y C X
< This is an immediate consequence of the definition. >
X
XX X
It can be easily seen that each intervally complete quasi-normed lattice is an order dense ideal of its own metric completion and every order ideal of any quasi-Banach lattice is an intervally complete quasi-normed lattice. Thus, the class of intervally complete quasi-normed lattices coincides with the class of order dense ideals of quasi-Banach lattices.
Lemma 3.5. Intervally complete quasi-normed lattice is uniformly complete.
< Let (xn) be a uniformly Cauchy sequence, that is, there exist e G X+ and a sequence of reals (en) such that limn en = ^d |xn+k — xn| ^ ene for all n, k G N. Then ||xn+k — xn|| ^
|e| and (xn) is Cauchy in (X, || ■ ||). Moreover, |xk+1| ^ |x1| + e1 e for all k G N. By hypothesis, there exists x = limnxn in (X, || ■ ||). Passage to the limit in |xn+k — xn| ^ ene with k ^ to yields |x — xn| ^ ene for all n G N, whence X is uniformly complete. >
Lemma 3.6. Let X be the metric completion of an intervally complete quasi-normed lattice X. Then X is Dedekind complete if and only if so is X.
< If X is Dedekind complete then so is X, since X is an order dense ideal of X. Assume that a quasi-normed lattice X is intervally complete and Dedekind complete and prove X is Dedekind complete. It was proved by Veksler [21, 22] that an Archimedean vector lattice is Dedekind complete if and only if it is uniformly complete and has the projection property. By Lemma 3.5 it suffices to show that X has the projection property. Consider an element x G X and a band B in X and pick a sequence (xn) in X converging to x. Observe, that B := B n X is a band of X and B± = B± n X, since X is an order dense ideal in X. If n stands for the band projection in X onto B, then n' := Ix — n is the band projection onto BThe sequences (nxn) and (n'xn) are Cauchy, as so is (xn), hence they converge to some u G X and u' G X, respectively. Clearly, u G B, u' G Band x = u + u'. >
X
order bounded Cauchy sequence in X+ is quasi-norm convergent.
< The proof given in [23, Theorem 1.1] for normed lattices works in the quasi-normed setting. >
Lemma 3.8. Let X be a universally complete vector lattice and (xa)aeA an increasing net in X+. Then there exists a band projection n on X such that supa nxa exists in X, while for the complementary band projection n' := — n we have Nn'e = supa n'(xa A Ne) for all N G N and e G X+.
< There is no loss of generality in assuming that X = CX(Q) with extremally compact space Q. (Recall that the symbol CX(Q) denotes the universally complete vector lattice of all continuous functions f : Q ^ [—to, to] for which the open set {q G Q : —to < f (q) < to} is dense in ^^^et (xa) ^e an increasing net in CX(Q) and define two functions x, x : Q ^ [0, to] by
x(q) = sup{xa(q) : a G A} (q G Q), x(q):= inf sup x(q') (q G Q),
UeN(q) q/eu
where N(q) is a basis of neighborhoods of q. Then x is lower semicontinuous and x is
continuous, see [24, Lemma V.1.2 and Theorem V.l.l], Consider an open set Qo := {q G Q : x(q) < to} and observe that its closure Q0 is clopen. Now, let n stands for the band projection of CX(Q) corresponding to Q0 and nx stands for the function coinciding with x on Q0 and vanishing on Q1 := Q \ Q0. Evidently, nx G CX(Q) and nx = supa nxa, see [24, Theorem V.2.1], At the same time x(q) = to for all q G Q1; so th at x(q) = to for all q G Q1 \ A where A is a meager subset of Q^. The latter implies that Ne(q) = supa xa(q) A Ne(q) for all q G Q1 \ A whence the desired equation Nn'e = supa n' (xa A Ne) follows. >
Lemma 3.9. Let X be a quasi-normed lattice X with the weak a-Fatou property. If X is intervally complete and Dedekind complete, then its maximal quasi-normed extension XK is intervally complete.
< Take an increasing order bounded Cauchy sequence (xn) in X+. Since XK is Dedekind complete, there exists x = supn xn. Prove that (xn) converges to x.
We may assume without loss of generality that A:= — xn||K<to.
Applying Lemma 3.8 to the increasing sequence (Zn) with zn := k(Xk+1 — Xk) yields
a band projection n on Xu such th at z := supn nzn exists in Xu and fo r n' := IXu — n we have Nn'e = supn n'(Zn A Ne) for all N G N and e G X 0 ^ e ^ z. Making use of the weak a-Fatou property and monotonicity of the quasi-norm we deduce
N||n'e|| ^ K supm ||n'(Zm A Ne)|| ^ K supm ||zm||K ^ KA < to.
It follows that n'e = 0 for all e G X and hence n'Z = 0 since X is order dense ideal in Xu. Thus, n = u and Z = supn zn G Xu. To ensure that z G XK it suffices to check that ||x|| ^ A for an arbitrary element x G X with 0 ^ x ^ z. For any such x put := zn A x and observe that (yn) is an increasing sequence in X+ with x = supn yn. Moreover, (yn) is Cauchy, since for arbitrary n, l G N we can estimate:
||yn+i — = ||Zn+i A x — z„ A x||k ^ ||zn+i — z„|k
= £ Ckk|xfe+i — xfc||k < E Ckk|xfc+i — xfc||k ^ 0
k=n+1 k=n+1
as n ^ to The interval comleteness of X implies that the sequence (yn) is convergent in X so that limn = supn = x % Lemma 2.6. Observe now that ||x|| ^ A, since ||yn|| ^ ||Zn|U ^ A and ||x|| = limn ||yn|| ^ A, whence Z G X
Now we are able to show that (xn) converges to x. First note that x — xn = o-^fc=n(xk+1 — xk), and consequently
n(x — xn) ^ o-^ k(xk+1 — xk) ^ z.
k=n
It follows that 0 ^
x xZn
^ (1/n)Z and ||
x xn11 k
^ (1/n)||Z||x ^ 0. Appealing to Lemma 3.7
completes the proof. >
Theorem 3.10. Let (X, || ■ ||X) be a Dedekind complete quasi-normed lattice with the weak a-Fatou property. The maximal quasi-normed extension (XK, || ■ ||K) is a quasi-Banach X
< The necessity is immediate from the fact that X is an order dense ideal of XTo prove the sufficiency observe that the metric completion (Y, || ■ ||y) of (XK, || ■ ||K) is Dedekind complete by Lemma 3.6. At the same time XK is order dense ideal of Y, since XK is intervally complete by Lemma 3.9 and an intervally complete quasi-normed lattice is an order dense ideal of its metric completion. Thus, X C Y C (XK)u = Xu and ||x|| = ||x||y for all x G X so that Y C XK by Lemma 3.3. It follows that Y = XK and XK is complete. >
It is evident that if X has the Levi property then X = XK but the converse is false, see [1, Examples 2 and 5]. The next result asserts that the maximal quasi-normed extension with the weak Fatou property has the Levi property.
X
quasi-normed extension XK has the Levi property if an d only if X has the weak Fatou property.
< Let X be a Dedekind complete quasi-normed lattice with the weak Fatou constant K. Take an increasing net (xa) in XK with B := supa ||xa||K < to. By Lemma 3.8 there exists a band projection n on XK such th at x = supa nza exists in Xu and for every N G N and
e G X+ we have Nn1e = supa nx(xa A Ne). Making use of the weak Fatou property we deduce N|n1e| ^ K supa ||n1(xa A Ne)|| ^ Ksupa ||xa||K = BK and n1 = 0 sinee ^aid e are arbitrary. It follows that n is the identity operator and x = supa xa. Show that x G X If x G X and 0 ^ x ^ x then xAxa G ^aid (x A xa) is an increasing net with the supremum x. By the weak Fatou property we have ||x|| ^ K supa ||xAxa|| ^ Ksupa ||xa||K = KB. It follows that sup{||x|| : x G X, 0 ^ x ^ x} ^ KB and x G X
To prove the converse, it suffices to observe that if XK has the Levi property, then X has the weak Fatou property by Proposition 2.12. >
X
ximal quasi-normed extension XK has the Fatou property if only if X has the Fatou property.
< The necessity is obvious. To prove the sufficiency take an increasing net (xa) in X+ such that x = supa xa for some x G X+. Pick an arbitrary x G X+ with 0 ^ x ^ x and note that xa A x is an increasing set in X+ and supa xa A x = x. In virtue of the Fatou property we have ||x|| = supa ||xa A x|| ^ supa ||xa||K. Hence, ||x|| ^ supa ||xa||K ^ ||x||K for all x G X+ with 0 ^ x ^ x. The latter implies that ||x||K = supa ||xa||K. >
X
property then the maximal quasi-normed extension XK has the Fatou and the Levi property.
< The proof follows immediately from Theorem 3.11 and Proposition 3.12. >
Remark 4.1. The maximal normed extension of a Dedekind complete normed lattice was introduced and the Theorem 3.10 was proved in Abramovich [1, Definition on p. 8 and Theorem 3]. Lemmas 3.6 and 3.7 for normed lattices can be seen in Veksler [22, Lemma 2] and [23, Theorem 1.1], respectively.
Remark 4.2. In the case of normed lattices Theorem 3.10 is true without the weak a-Fatou property, see Abramovich [1]. We do not know whether or not the assumption about the weak a-Fatou property is superfluous in Theorem 3.10.
Remark 4.3. Let X be et CjlletSl- Banach lattice and (fi,Rloc,^) a vector measure space with a localizable measure ^ : R ^ X+ which is countable additive in the sense of order or quasi-norm convergence depending on the context, see [10, 11]. The Bartle-Dunford-Schwartz type integration and the purely order based Kantorovich-Wright integration with respect to H provide two quasi-Banach lattices of integrable functions, L^ and L,^), respectively, see [10]. Moreover, the vector lattice L°(^) (of equivalence classes) of ^-a.e. finite Rloc-measurable real-valued functions is a universal completion of both quasi-Banach lattices L^ and L^ According to Definition 3.2 we can construct maximal quasi-normed extensions (L¿K||-||oK) and (L^^|| ■ ||tK) of and L^respectively. By virtue of Theorem 3.10 L^^is a
quasi-Banach lattice and an order dense ideal in L°(^). Moreover, L^^has the Fatou and Levi properties by Corollary 3.13, since L^is order continuous.
Remark 4.4. Similarly, is a quasi-normed lattice and order dense ideal in L°(^),
but is metrically complete under the additional assumption that has the weak
a-Fatou property. We do not know whether is metrically complete (and hence a quasi-
Banach lattice) without this additional assumption coming from Theorem 3.10.
Definition 4.5. An Rloc-measurable function f : fi ^ R u |±to>} is called weakly integrable with respect to ^ or weakly ^-integrable if
4. Concluding remarks
where |x*^| : Rloc ^ [0, to] variation of x*^ and B* the positive part of the unit ball in X*. A weakly integrable function f is integrable with respect to ^ if for each A G Rloc there exists a vector denoted by JA f d^ G X, such that
x*( J f^) = /f — x* G X*.
aa
Denote by L^(^) the space of (equivalence classes) all weakly ^-integrable function equipped with the norm || ■ and 1 et L1 stand for the subspace of consisting of (equivalence
classes) all ^-integrable functions. Note that if ||f ||m < to then |f | < to ^-a.e. Thus, L^ and L1 can be considered as subspaces of L0(^).
Theorem 4.6. Let X be a Banach lattice and (fi, Ra vector measure space with R-decomposable me asure ^ : R ^ X+. Then L^ and L^H coincide as Banach lattices.
Remark 4.7. In Theorem 4.6 R-decomposability of measure ^ provides L^(^) with the Levi and Fatou properties (see [4, Theorem 5.8]), while L^^always has these properties. Without R-decomposability assumption it may happen that L^ = L^^Similar questions for the space of order integrable functions L,^) and the corresponding maximal quasi-Banach extension remain open.
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Received 14 July, 2011
Kusraev Anatoly Georgievich
Vladikavkaz Science Center of the RAS, Chairman
22 Markus Street, Vladikavkaz, 362027, Russia;
North Ossetian State University,
Head of the Department of Mathematical Analysis
44-46 Vatutin Street, Vladikavkaz, 362025, Russia
E-mail: [email protected]
Tasoev Batradz Botazovich Southern Mathematical Institute — the Affiliate of Vladikavkaz Science Center of the RAS, Researcher 22 Markus street, Vladikavkaz, 362027, Russia E-mail: [email protected]
О МАКСИМАЛЬНОМ КВАЗИНОРМИРОВАННОМ РАСШИРЕНИИ КВАЗИНОРМИРОВАННЫХ ВЕКТОРНЫХ РЕШЕТОК
Кусраев А. Г., Тасоев Б. Б.
Цель работы - распространить конструкцию Абрамовича максимального нормированного расширения нормированной решетки на класс квазинормированных решеток. Установлено, что максимальное квазинормированное расширение Xк норядково полной квазинормированной решетки X со слабым счетным свойством Фату является квазибанаховой решеткой в том и только в том случае, когда X интервально полна. Боле того, Xк обладает свойствами Леви и Фату, если только X — порядково полная квазинормированная решетка со свойством Фату. Обсуждается также возможность применения этой конструкции к определению пространства слабо интегрируемых функций относительно меры со значениями в квазибанаховой решетке, не прибегая к двойственности (которая может оказаться тривиальной).
Ключевые слова: квазинормированная решетка, максимальное квазинормированное расширение, свойство Фату, свойство Леви, векторная мера, слабо интегрируемые функции.
