Internat. J. Math & Math Sci.
(1986) 17-22
Vol. 9 No.
17
FUNCTORIAL PROPERTIES OF THE LATTICE
OF FUNCTIONAL SEMI-NORMS
I. E. SCHOCHETMAN and S. K. TSUI
Oakland University
Rochester, Michigan 48063
(Received August I0, 1984)
ABSTRACT.
Given a measureable transformation between measure spaces, we determine
when such gives rise to a mapping between the corresponding lattice of function
We further determine when this mappings preserves norms and observe that
semi-norms.
it does preserve certain other important properties.
connection between measure spaces and lattice.
We next establish a functorial
Finally, we show that the above
lattice mapping does not commute with the associate construction.
KEY WORDS AND PHRASES:
Function semi-norm, associate semi-norm, lattice
of semi-norms,
masure-preservin.g transformation, semi-norm preserving, associate preserving, lattice
subhomomorphism, category,, fnctor.
1980 AMS SUBJECT LASIFICATION CODE. 18A20, 18B99, 28A65, 06A20, 18A99, 18D35.
I.
INTRODUCTION.
Let (X, S, ) be a sigma-finite measure space and
-measurable functions on X.
M+()
the space of [0,=]-valued
Contrary to conventional practice, it will not be con-
M+()
M+().
-a.e.
venient to identify two functions in
which are equal
Z() denote the -null function in
Thus, Z() is the null equivalence class in
M+{)
of the zero function on X.
a mapping
O:M+()/[0, =]
Accordingly, let
In this setting, a (function) semi-norm on
having the following properties.
Let c>0, and f,g
M+() is
M+().
Then:
(I)
f-g
(2)
f e Z() implies 0(f)
(3)
0(cf)
(4)
o(f+g)
(5)
fg -a.e. implies o(f)J 0(g).
Z() implies 0(f)
0(g).
0.
c0(f).
0(f)+ 0(g).
The semi-norm 0(f)
0 implies feZ().
Let P() denote the set of all semi-norms and
P () the subset of all norms (never empty).
O
Observe that P() is canonically partially ordered by:
Ol
02 if 01(f)
02(f),
faM+().
I. E. SCHOCHETMAN AND S. K. TSUI
18
It is well-known that, relative to this ordering, P(B) is a complete lattice with sup
and inf given by
(f),
sup(01
(01
v
02)
(f)
(01
A
02)
(f)= inf
02
(f)),
and
+ 02
{01 (fl)
M+()’ fl+f2 =f’-a’e’}
(f2): fl’ f2
(See sections 3 and 4 of [3] for the sup and inf of arbitrary families in P().)
Now let (Y,T,v) be another sigma-finite measure space and :X+y a measurable
transformation.
For such
,
This in turn yields a mapping
M+(v).
In general, (0)
:0+0
O
O(g)
g.
Moreover, if 0 is a norm, then (0)
Thus, the first question we ask is:
Under
In section 2, we give necessary and suf-
semi-norm-preserving?
ficient conditions for this to be the case (2.2).
Under what
The next question is:
In section 3, we give necessary and suf-
norm-preserving?
additional conditions is
defined by
from P(H) into the [0,=]-valued functions on
is not a semi-norm.
may be a semi-norm which is not a norm.
what conditions is
:M+()+M+()
we obtain a mapping
ficient conditions for this to be the case (3.5).
There are certain very important
sublattices in the lattice of semi-norms which have been studied extensively (see [2,3]).
Also in section 3, we observe that all of these sublattices are preserved by
when
The previous results suggest there is a functorial
is semi-norm-preserving.
However, when
connection between measure spaces and lattices.
Specifically, in general,
may not be a lattice homorphism.
’s".
equal the infimum of the
(4.3).
(3.7)
is semi-norm-preserving,
"
of an infimum does not
is a lattice "subhomomorphism"
Despite this failing,
With this notion of lattice morphism, we are able (in section 4) to establish
Finally, in section 5, we see that the mapping
the desired functorial connection.
and the assocconstriuction 0+0
For this purpose, recall
are incompatible in general.
that
o’(f)
sup
{fX
fgd:o(g)
i}, f
M+(B)
Also, let N(B) denote the space of B-null subsets of X (similarly for
2.
SEMI-NORM PRESERVATION.
Before investigating the conditions under which
preserves semi-norms, let us
see first that it does not have this property in general.
{a,b}
Y
2.1 Example. Let X
O. Let
and v({b})
v({a})
{b}
-l({b})
D(f)
N().
lflll
with
Let 0 be the
(Z(v))
Z(B).
null equivalence classes in
2.2 Theorem.
(i) :P()
(ii)
-l(N(v))
(iii)
(Z(D))
Ll-norm
f(a) + f(b), f
The function g on Y defined by g(a)
B-null, i.e.
and v defined as follows:
be the identity mapping.
Thus, (0)(g)
The following are equivalent:
({b})
N(v), while
i.e.,
M+(B)
O, g(b)
M+(v).
Po(B),
in
({a})
Then {b}
I, is v-null.
However,
g
is not
0(g) # O, i.e. (0) is not constant on
FUNCTORIAL PROPERTIES OF THE LATTICE OF FUNCTION SEMI-NORMS
Proof.
Let gl’ g2
(ii) implies (i):
{x
i
gl (x)
X:
E
M+(v)
E
g2’ v-a.e.
g2(Y )})"
be such that gl
-1({y
g2 (x)}
y
gl(y
i
19
Then
Since the set in the right parentheses is v-null, it follows from (ii) that its inverse
is -null, i.e.
image under
gi
0(gl)
(0)(gl)
g2
,
0(g2)
i.e. (0) satisfies (5) of i.
(0)(g2),
(XE)
O(Xe) X-I(E)
_I(E)
Z(), i.e.
E
(i) implies (iii):
-I(E)
Let g
0(f)
g
0.
The characteristic function X E is
We then have
e N().
Suppose (iii) is false.
f is not in Z().
have (0)(g)
Z() by (iii).
e
Then there exists f in
Z(v) be such that f
g.
If 0
2.3 Remarks.
Observe that (iii) of the theorem says that
M+()
to the zero-class in
Specifically, if f e Z(), then
2.4 Definition.
ePo(),
(Z(v))
such that
then by (i) we must
This contradicts the fact that 0 is a norm.
zero-class in
Z().
The remaining properties
Therefore, (0) e P(v), for all 0 e P().
Let E be an element of N(v).
then in Z(v), i.e.
so that
(I) of I.
This also proves
(2), (3), (4) of I are easily verified.
(iii) implies (ii):
XE
Hence, for 0 e P(), we have
-a.e.
M+()
O(g)
o
essentially sends the
because, modulo nullity,
(Z(v))
f, -a.e., for g the zero function on Y.
The measurable transformation
:X
Y is semi-norm-preserving if the
conditions of 2.2 hold.
3.
PROPERTY PRESERVATION.
The natural next question to ask about
does it preserve norms?
some measure
is the following:
Under what conditions
The answer to this question is somewhat complicated because of
theoretic technicalities.
are necessitated by the fact that
These (together with some additional notation)
need not preserve measurable sets, i.e. it may not
be bimeasurable.
Let
denote the completion of
and
N()
N()
In general,
{E
-I((E))
N():
The transformation
and only if
-l(N())
f
K(0)
Of course, K(0)
3.2 Lemma.
N()) if
-I(N())
automatically have the extra property.
0(f)
0}.
Z() in general.
is semi-norm preserving.
Suppose
K((0))
Proof.
-l(N(v))
0, define
M+():
{f
However:
is semi-norm-preserving, (i.e.
N().
The elements of
For any semi-norm
Let v* (resp. v,) denote
Also let
E}.
is a proper subset of N().
3.1Lemma.
Proof.
.
its domain [i].
the outer (resp. inner) measure derived from
(o)-I
If 0 e P(), then
(K(0)).
Straightforward.
We then have the following answer to our question:
3.3 Proposition.
is a norm if and only if
Proof.
is semi-norm preserving and 0 is a norm in P(B).
Suppose
()-l(z())
Then (0)
Z(v).
Apply 2.2 and 3.2.
In order to obtain an answer analogous to 2.2 in terms of
require the following.
itself, we first
20
I. E. SCHOCHETMAN AND S. K. TSUI
(X) (set difference).
Y
Let C
3.4 Lemma.
P(V) and (0) is a norm, then
If 0
v,(C)
0, i.e.
Proof.
If not, there exists E in T such that E
C and (E) > O.
we have
g
Z().
is ,-essentially onto.
Z(v), so that (0)(g)
(X)
(i)
(0) is a norm.
N
(recall 2.2, 2.3).
Z(v)
N(),
Let E
F
(E)
(Z())
Z(), i.e. (F)
0.
implies (i):
(0 is a norm).
(E)
-I((E))
0 and
E.
Then R E
(F)
M+(v)
Let g
so that
T be such that F i (E). Then F
(E) and
Z(), i.e.
Z(). This implies that F
(E) E" Let
E" Hence, F
it
0.
(i) is equivalent to (iii) by 3.3.
(iii) implies (ii):
Z() and
v,(C)
-I(N(,)).
()
()-l(z())
(iii)
Proof.
E’
T, then the following are equivalent:
If
(ii)
Then for g
is semi-norm preserving, 0 is a norm inP() and
Suppose
3.5 Theorem.
0, while g
Consequently, v,((E))
be such that (o)(g)
0.
O, so that (E)
Then o(g)
F !
(o)-I
N(v).
0, so that
(li)
g
Z()
Since
-l(supp(g)) supp(g),
0,
follows that (-l(supp(g)))
-l(supp(g))
i.e.
supp(g) n (X), G c
supp(g) n C,
C
Then G r, G c
T.
and
to
belong
observing that (X)
N().
Let
Gr
T, supp(g)
Gr
tJ
G c (disjoint)
and
-l(Gr) -I(N(,)),
-l(supp(g))
v,(G r)
,(G c)
by condition (ii), so that
(G c)
(G c)
O.
0
On the other hand,
[I, p. 60].
Therefore,
(G c)
(supp(g)) ! ,(G r) +
so that g
Z(), i.e. (0) is a norm.
3.6 Corollary.
Suppose
0
[I, p. 61],
is semi-norm-preserving and 0 is an norm in
P(V).
If
is
bimeasurable and maps X v-essentially onto Y, then (i) and (iii) of the theorem are
equivalent to (ii’):
N(V)
-I(N()).
We next consider our question in the context of the subsets of P(V) introduced in
section 2 of [3]. Here the answers are the best possible. The subsets consist of
those norms having the Riesz-Fisher (R), weak (W) or strong (S) Fatou property, those
satisfying the infinite triangle inequality (I) and those which are of absolutely continuous norm (A)
3.7 Theorem.
(see[2,3,4]).
For the following, let B denote either R,I,W, S or A.
norm-preserving, then
Proof.
preserves the property defining B, i.e.
If
in proving the theorem for the case B
is semi-
B()
The proof for each choice of B is more-or-less straightforward.
we leave the details to the interested reader
4.
:
B().
Therefore,
after remarking that 3.2 is required
R.
THE FUNCTOR.
In this
section we investigate the categorical connection between measurable
transformations and lattices of semi-norms.
norm-preserving, the corresponding morphism
As the next example shows, if
is semi-
may not be a lettice homomorphism.
FUNCTORIAL PROPERTIES OF THE LATTICE OF FUNCTION SEM-NORMS
4.1 Example.
Let X
Define (x)
y, x
and we have
:
N the set of all positive integers, and Y
X.
g
P()
{y}
I.
with (Y)
is a semi-norm-preserving measurable transformation
P() as in Section 2.
Z
01(f
Then
21
Define
f(n)/2 n
and
02(f)
lim sup(f(n))
n
Let g be the function equal to
+
sup(f(n)), f
E
n
on Y, so that g
M+().
M+().
We leave to the reader the
verification that
[(0 I) A
(02)
(g)
While
Thus,
[(Pl A p2) (g) _<
(Pl A 02) # (pi) A (p2
Despite this failing,
4.2 Lemma.
A
.
I,
(p2)
If
01, 02
in general.
does have suitable lattice morphism properties.
e P(), then
(Pl
V
(01)
02
V
(p2
and
(01
A
02) _< (01)
in general.
4.3 Definition.
Any mapping between lattices having the properties exhibited by
in
4.2 will be called a lattice subhomomorphism.
We are now ready to define a functor.
On the one hand, consider all sigma-finite
measure spaces as the objects and semi-norm-preserving, measurable transformations as
the morphisms.
These form a category which we denote by X.
On the other hand, con-
sider all lattices as the objects and lattice subhomomorphisms as the morphisms.
form a category which we denote by P.
F
These
By the results of section 2, we obtain a "mapping"
X/P
determined by
F(X, S, )
and
F()
P(), (X, S, ) e Obj (X),
,
Mor((X, S, ), (Y, T, )).
We leave to the reader the task of verifying the F is in fact a functor.
5. ASSOCIATE PRESERVATION.
Our final concern is the question of whether
see
preserves associates.
We shall
in the next examples that (0’) and (0)’ are not even comparable in general.
Let X,Y,, be as in 4.1.
5.1 Example.
of X,Y by f,g.
Z
0(h)
Then 0(f)
Denote the respective characteristic functions
Let 0 be the norm in P() given by
h(n)/2 n,
h
M+().
and
(0)’(g)
sup{lh(y)
sup{lh(y)
0(h) < i}
h(y)o(f) < I}
0(f) -I
I.
On the other hand,
(0’)(g)
Thus, (0’)
i
sup {Z
lh(n)
(0)’, in general.
0(h)
I}
I. E. SCHOCHETMAN AND S. K. TSUI
22
Now let X
5.2 Example.
semi-norm-preserving
g(y)
{x}, Y
I.
with (X)
I, so that
is
Let h denote the characteristic function of Y and define
Define (x)
0, y--
Let 0 be the norminP() given by 0(f)
(0’)(g) --g(1)
sup{If(1)
f(x).
Then
o(f) < I}
0.
On the other hand,
If(n) Ig(n)
E2 If(n)
(0)’(g) --sup{E
-<
Thus, (P’)
5.3 Remarks.
i
P(E)
_<
I}
(0)’, in general
It is possible to find non-trivlal conditions on
guarantee a comparison of (0’) and (0)’.
are not far from requiring that
essentially one-one)
which will at least
However, the conditions we have in mind
be an essential measure isomorphism (need not be
Thus, the strength of the hypothesis, combined with the weak-
ness of the conclusion (namely,
(0’)
(P)’), provide little motivation for present-
ing the details here.
REFERENCES
I.
HALMOS, P.R.
2.
LUXEMBURG W.A.J. and ZAANEN, A.C.
Math. 25 (1963), 135-153
3.
SCHOCHETMAN, I.E. and TSUI S.K. On the lattice theory of function semi-norms,
Internat. J. Math. & Math. Sci. 6 (1983), 431-447.
4.
ZAANEN, A.C.
Measure Theory, Van Nostrand, Princeton, 1962.
Notes on Banach function spaces I, II, Indag.
ntegration, North Holland, Amsterdam, 1967.