Table Of ContentSOME MEASURE-THEORETIC PROPERTIES OF
GENERALIZED MEANS
5
1
IRINANAVROTSKAYAANDPATRICKJ.RABIER
0
2
n
a
J Abstract. IfΛisameasurespace,u:Λm→RisagivenfunctionandN ≥
2 N −1
1 m, the function U(x1,...,xN) = (cid:18) m (cid:19) 1≤i1<···<im≤Nu(xi1,...,xim)
P
iscalledthegeneralizedN-meanwithkernelu,aterminologyborrowedfrom
] U-statistics. Physical potentials for systems of particles are also defined by
A generalizedmeans.
F Thispaperinvestigateswhethervariousmeasure-theoreticconceptsforgen-
eralized N-means are equivalent to the analogous concepts for their kernels:
.
h a.e. convergence of sequences, measurability, essential boundedness and in-
t tegrability with respect to absolutely continuous probability measures. The
a
answer isoften, but not always, positive. This informationiscrucial insome
m
problems addressing the existence of generalized means satisfying given con-
[ ditions, such as the classical Inverse Problem of statistical physics (in the
canonical ensemble).
1
v
1. Introduction
0
5 Let Λ be a set and let g :Λm →R be a given function. If N ≥m is an integer,
8
the generalized N-mean of g is the symmetric function U : ΛN → R defined by
2
0 U(x ,...,x ) := (N−m)! g(x ,...,x ), where the sum carries over all m-tuples
1 N N! i1 im
1. (i1,...,im)∈{1,...,N}mPsuch that ij 6=ik if j 6=k.
0 If u(x1,...,xm) := m1! g(xσ(1),...,xσ(m)) where σ runs over the permutations
5 of {1,...,m}, then u is syPmmetric and
1
:
v −1
N
Xi (1.1) U(x1,...,xN)=(cid:18) m (cid:19) u(xi1,...,xim).
1≤i1<X···<im≤N
r
a The function u is customarilycalled the kernelofU andm is the order(or degree)
of U. Obviously, u = g if m = 1 but, unlike g, the kernel u is always uniquely
determined by U, a point to which we shall return below.
Generalized means lie at the foundation of U-statistics, a field extensively stud-
ied since its introduction by Hoeffding [4]. The nomenclature (generalized mean,
kernel, order) is taken from it. In U -statistics, the kernel u is given, the variables
x ,...,x are replaced with random variables X ,...,X and the interest centers
1 N 1 N
on the behavior of statistically relevant quantities as N →∞.
PhysicalpotentialsforsystemsofN particleswhentheinteractionofanym<N
particles is taken into account are also defined by generalized means (without the
scaling factor, an immaterial difference). The variable x captures the relevant
j
1991 Mathematics Subject Classification. 28A20,28A35,82B21,94A17.
Key words and phrases. Generalizedmean,kernel,measurability,convergence, U-statistics.
1
2 IRINANAVROTSKAYAANDPATRICKJ.RABIER
information about the jth particle (often more than just its space position, so
that Λ need not be euclidean space) and the energy u(x ,...,x ) depends upon
i1 im
the interaction of the particles i ,...,i . In some important questions, notably the
1 m
famous“InverseProblem”ofstatisticalphysics([1],[2],[7]),thepotentialU isnot
given. Instead,theissueispreciselythe existenceofapotentialU oftheform(1.1)
satisfying given conditions.
Forexample,whentheInverseProblemissetupinthe“canonicalensemble”,N
is fixed1 and U should maximize a functional F(V) (relative entropy) over a class
of potentials V of the form (1.1) characterized by various integrability conditions.
Accordingly,the setΛis equipped witha measuredxanda generalizedN-meanU
of order m is still defined by (1.1), but now with equality holding only a.e. on ΛN
for the product measure. If a maximizing sequence U is shown to have some type
n
of limit U, is U a generalizedN-mean of order m? In the affirmative,what are the
properties of the kernel u of U that can be inferred from the properties of U ?
Inthispaper,weprovideanswerstotheabovequerieswhich,apparently,cannot
befoundelsewhere. Theseanswersplayakeyroleintherecentwork[7]bythefirst
author and they should have value in the broad existence question for generalized
means. The existence of extremal potentials is essential in particle physics; see for
instance [8] for a variant of the Inverse Problem arising in coarse-grain modeling,
butusuallyassumed(ifnottakenforgranted). Moregenerally,therearenumerous
examples borrowing from information theory when generalized means are sought
that maximize some kind of entropy. Even though the maximization involves only
a finite number of parameters in many of these examples, it should be expected
that some of the more complex models require a measure-theoretic setting.
Fromnowon,themeasuredxonΛisσ-finiteandcompleteand,toavoidtrivial-
ities, Λ has strictly positive dx measure. The σ-finiteness assumption ensures that
the product measure dx⊗k on Λk and its completion dkx are defined and that the
classical theorems (Fubini, Tonelli) are applicable. The terminology “a.e.”, “null
set”or“co-nullsubset”(complementofanullset)alwaysreferstothemeasuredkx
or its measurable subsets. Since dkx is complete and Λk has positive dkx measure,
a co-null subset of Λk is always measurable and nonempty.
The symmetry of u or U is not needed in any of our results and will not be
assumed. Accordingly,the“generalizedmeanoperator”G givenbyG (u):=
m,N m,N
U iswelldefined(Remark2.1)andlinearonthe spaceofalla.e. finitefunctions on
Λm. The expositionis confinedto real-valuedfunctions, but everythingcanreadily
be extended to the vector-valued case.
The recovery of the kernel u from the generalized mean U will play a crucial
role. There are elementary ways to proceed, which may explain why no general
procedure seems to be on record. (Lenth [6] addresses the completely different
problem of finding u when U = U in (1.1) is known for every N but the order
N
m is unknown.) However, the problem becomes much more subtle after noticing
that the most natural formulas for the kernel are useless for measure-theoretic
purposes. Indeed, after suitable modifications of u and U on null sets, it may be
assumed that (1.1) holds pointwise. Then, if m = 1, we get u(x) = U(x,...,x)
but, of course, this does not show whether the measurability of U implies the
measurability of u. When m>1, other simple formulas for the kernel, for instance
1Amajordifficulty; inthe“grandcanonical ensemble”,N isfreeandtheproblemwassolved
byChayesandChayes [1].
GENERALIZED MEANS 3
u(x ,x )= NU(x ,x ,...,x )−(N−2)U(x ,...,x )whenm=2,sufferfromsimilar
1 2 2 1 2 2 2 2 2
shortcomings.
More sophisticated representations of u in terms of U will be needed to prove
that,indeed, uismeasurablewheneverU is measurable(Theorem3.1). Theserep-
resentationscallfortheexplicitintroductionofthe“kerneloperator”K inverse
m,N
of G , i.e., K (U) = u. We shall show that, when N > m ≥ 2, K (U)
m,N m,N m,N
can be recovered from U and the four operators K ,K ,G and
m,N−1 m−1,N N−1,N
G (see (3.3) and also (3.1) when m =1; that K =I is trivial). It fol-
m−1,N−1 m,m
lowsthatmanypropertiesofthekernelscanbeestablishedbytransfiniteinduction
on the pairs (m,N) with m ≤ N, totally ordered by (m,N) < (m′,N′) if m < m′
orm=m′ andN <N′. This methodis systematicallyusedthroughoutthe paper.
There are also unexpected differences between pointwise and a.e. limits of gen-
eralized N-means. Suppose once again that m = 1 and that U (x ,...,x ) =
n 1 N
N−1 N u (x ) for every (x ,...,x ) ∈ ΛN, so that u (x) = U (x,...,x). If
i=1 n i 1 N n n
the sPequence Un has a pointwise limit U, then un has the pointwise limit u(x) =
U(x,...,x). This is true irrespective of whether U achieves infinite values on ΛN.
In contrast, if it is only assumed that U has an a.e. limit U, the sequence u (x)
n n
mayhaveno limit for any x∈Λ;see Example 2.1. Nonetheless,we shallsee in the
next section that if U is a.e. finite, then u (x) converges for a.e. x∈ Λ (Theorem
n
2.3). Thus, the finiteness of the limit U, irrelevant when pointwise convergence is
assumed 2, makes a crucial difference when only a.e. convergence holds.
Measurability and essential boundedness are discussed in Section 3. Section 4
is devoted to more delicate integrability issues. The general problem is as follows:
Asymmetricprobabilitydensity P onΛN induces a naturalsymmetric probability
densityP onΛmwithm≤N uponintegratingP withrespecttoanysetofN−m
(m)
variables. Is it true that a generalized N-mean U of order m is in Lr(ΛN;PdNx)
if and only if its kernel u is in Lr(Λm;P dmx) ?
(m)
Assuming P >0 a.e., the answer is positive when r =∞, but the necessity may
be false if r <∞ (Example 4.1). This has immediate and important consequences
in existence questions. In a nutshell, the problem of finding a generalized N-mean
in Lr(ΛN;PdNx) is not always reducible to the problem of finding its kernel in
Lr(Λm;P dmx) and the former may have solutions when the latter does not.
(m)
The next step is to investigate whether conditions on P ensure that the above
discrepancies do not occur. Such a condition is given in Theorem 4.5. It always
holds in some arbitrarily small perturbations of any symmetric probability density
(Theorem 4.7) and, when Λ has finite dx measure, it is even generic (in the sense
of Baire category)among bounded symmetric probability densities (Theorem 4.8).
Thissupportstheideathat,whileprobablynotnecessary,theconditioninquestion
is sharp. Yet, it is instructive that it fails when PdNx is the probability that
N particles have given coordinates, under the natural assumption that P = 0
when two coordinates are equal. If so, the existence of potentials U = G (u) ∈
m,N
Lr(ΛN;PdNx) with u∈/ Lr(Λm;P dmx) cannot be ruled out.
(m)
Our approach also provides good guidelines for the treatment of the more gen-
eral problem when (1.1) is a weighted mean, but since there are significant new
technicalities, exceptional cases, etc., this problem is not discussed.
2If m > 1, the finiteness of U is not entirely irrelevant to the pointwise convergence issue,
whichhoweverremainsmarkedlydifferentfroma.e. convergence.
4 IRINANAVROTSKAYAANDPATRICKJ.RABIER
2. Almost everywhere convergence
In this section, we prove that a sequence U of generalized N-means of order m
n
has an a.e. finite limit U if and only if the corresponding sequence of kernels u
n
has an a.e. finite limit u. The next example shows that the “only if” part is false
if U is infinite.
Example 2.1. With Λ = [0,1] and dx the Lebesgue measure, let f denote the
n
well-known sequence of characteristic functions of subintervals J with |J | → 0,
n n
such that, when x is fixed, f (x) assumes both the values 0 and 1 for arbitrarily
n
large n ([3,p. 94]). If u :=2n(1−f ), then u =0 on J and u =2n otherwise.
n n n n n
Also, u (x) has nolimit for any xsinceu (x) assumes both thevalues 0and 2nfor
n n
arbitrarily large n. On the other hand, U := G (u ) is 0 on J ×J and either
n 1,2 n n n
n or 2n otherwise. As a result, U tends to ∞ off the diagonal of [0,1]2 -a set of
n
d2x measure 0- since (x ,x ) ∈/ J ×J if x 6= x and n is large enough. Thus,
1 2 n n 1 2
U has an a.e. limit but u does not.
n n
If U = G (u ), then (a) u and U can be modified on n-independent null
n m,N n n n
setsofΛm andΛN,respectively,insuchawaythatu iseverywherefiniteandthat
n
U is given by (1.1) for every (x ,...,x ) ∈ ΛN (so that U is everywhere finite;
n 1 N n
see also Remark 2.1 below). On the other hand, (b) if U → U a.e., U and U
n n
canbemodifiedonnullsetsindependent ofninsuchawaythatconvergenceholds
everywhere. However, (a) and (b) cannot be achieved simultaneously. Indeed, a
modificationofthe left-handside of(1.1)ona nullsetof ΛN need notpreservethe
sum structure of the right-hand side, whereas such a modification may be needed
to ensure that U → U pointwise. Thus, it is generally not possible to assume
n
both pointwise convergence and pointwise sum structure of U . In fact, if this
n
were always possible, the sequence u (x) of Example 2.1 would have the a.e. limit
n
U(x,x), which is false for every x.
We begin with a simple lemma.
Lemma2.1. Let1≤k ≤N beintegers andlet T bea co-nullset ofΛk.Then, the
k
set T :={(x ,...,x )∈ΛN :(x ,...,x )∈T for every 1≤j <···<j ≤N}
N 1 N j1 jk k 1 k
is co-null in ΛN.
Proof. As a preamble, note that the product S ×T of two co-null sets S and T
in Λj and Λℓ, respectively, is co-null in the product Λj+ℓ since its complement
(Λj\S)×Λℓ ∪ Λj ×(Λℓ\T) is a null set in Λj+ℓ. This feature is immediately
e(cid:0)xtended to a(cid:1)ny (cid:0)finite product(cid:1)of co-null sets. Now, TN is the intersection of the
sets {(x ,...,x ) ∈ ΛN : (x ,...,x ) ∈ T } for fixed 1 ≤ j < ··· < j ≤ N. Such
1 N j1 jk k 1 k
a set is the product of N −k copies of Λ and one copy of T and therefore co-null
k
in ΛN from the above. Thus, T is co-null. (cid:3)
N
Remark 2.1. It follows at once from Lemma 2.1 that an a.e. modification of u
creates only an a.e. modification of G (u).
m,N
The next lemma is the special case m=1 of Theorem 2.3 below.
Lemma 2.2. Let U be a sequence of generalized N-means of order 1 and let u
n n
denote the corresponding sequence of kernels (i.e., U =G (u )). Then, there is
n 1,N n
an a.e. finite function U on ΛN such that U → U a.e. if and only if there is an
n
a.e. finite function u on Λ such that u →u a.e. on Λ. If so, U =G (u).
n 1,N
GENERALIZED MEANS 5
Proof. Assume first that u exists. With no loss of generality, we may also assume
that u and u are everywhere defined and that u → u pointwise on Λ. Then,
n n
U →G (u) pointwise.
n 1,N
Conversely,assumethatU exists. Withnolossofgenerality,wemayalsoassume
that u and U are everywhere finite and that U (x ,...,x )=N−1 N u (x )∈
n n 1 N i=1 n i
R for every (x1,...,xN)∈ΛN. By hypothesis, P
(2.1) U (x ,...,x )→U(x ,...,x ),
n 1 N 1 N
for a.e. (x ,...,x ) ∈ ΛN, but recall that it cannot be assumed that U → U
1 N n
pointwise; see the comments after Example 2.1.
Let E ⊂ ΛN denote the co-null set on which (2.1) holds. There is a co-null set
S of ΛN−1 such that, for every (x ,...,x ) ∈ S , the subset E :=
N−1 2 N N−1 x2,...,xN
t{ehxa∈t aΛ.e:.((xx,x,2..,..,..x,xN)-s)e∈ctEio}niosfctoh-enuclolmeinplΛem. TeenhtisoifsEmeiesrealynuallrespethroafsΛin.geof thee fact
2 N
e e
From now on, (x ,...,x ) ∈ S is chosen once and for all. By definition of
2 N N−1
e e
S , if x∈E , then
N−1 x2,...e,xN e e
(e2.2) eun(xe)+un(x2)+···+un(xN)→NU(x,x2,...,xN).
Thus, if (x ,...,x )∈EN , (2.2) implies
1 N x2,.e..,xN e e e
u (x )+u (x )e+··e·+u (x )→NU(x ,x ,...,x ),1≤j ≤N
n j n 2 n N j 2 N
and so, by addition,
e e e e
(2.3) u (x )+···+u (x )+N(u (x )+···+u (x ))→Nl(x ,...,x ),
n 1 n N n 2 n N 1 N
where l(x1,...,xN):=ΣNj=1U(xj,x2,...,xeN), i.e., e e
(2.4) e l =NGe1,N(Ue(·,x2,...,xN)).
Since both E and ExN2,...,exN are co-null ineΛN, teheir intersection E∩ExN2,...,xN is
co-null. Let (x1,...,xNe) ∈ Ee ∩ExN2,...,xN be chosen once and for all, so tehat e(2.1)
and (2.3) hold. This implies that u (x )+···+u (x ) has a finite limit, namely
e n e2 n N
l(x ,...,x )−U(x ,...,x ).Then,by(2.2)and(2.4),u (x)tendstothefinitelimit
1 N 1 N n
e e
e(2.5) u(x):=
NU(x,x ,...,x )+U(x ,...,x )−NG (U(·,x ,...,x ))(x ,...,x ),
2 N 1 N 1,N 2 N 1 N
for every x ∈ E . Thus, u → u a.e. on Λ and so, by the first part of the
xe2,...,xNe n e e
proof, U =G1,Ne(u).e (cid:3)
Theorem 2.3. Let U be a sequence of generalized N-means of order m ≥ 1 and
n
let u denote the corresponding sequence of kernels (i.e., U =G (u )). Then,
n n m,N n
there is an a.e. finite function U on ΛN such that U → U a.e. if and only if
n
there is an a.e. finite function u on Λm such that u → u a.e. on Λm. If so,
n
U =G (u).
m,N
Proof. AsintheproofofLemma2.2whenm=1,thesufficiencyisstraightforward.
We nowassumethat U exists. With nolossofgenerality,we mayalsoassumethat
u and U are everywhere finite and that
n
−1
N
(2.6) U (x ,...,x )= u (x ,...,x )∈R,
n 1 N (cid:18) m (cid:19) n i1 im
1≤i1<X···<im≤N
6 IRINANAVROTSKAYAANDPATRICKJ.RABIER
for every (x ,...,x )∈ΛN. By hypothesis,
1 N
(2.7) U (x ,...,x )→U(x ,...,x ),
n 1 N 1 N
for a.e. (x ,...,x ) ∈ ΛN. Once again, it cannot also be assumed that U → U
1 N n
pointwise.
The proofthat u has ana.e. finite limit u will proceedby transfinite induction
n
on the pairs (m,N) with N ≥ m, totally ordered by (m,N)< (m′,N′) if m < m′
or if m = m′ and N < N′. For the pairs (m,m) (with no predecessor), this must
be proved directly, but is trivial since u = U in this case. If m = 1, the result
n n
follows from Lemma 2.2 irrespective of N. Accordingly, we assume m ≥ 2 and
that the result is true for generalized N-means of order m−1 with N ≥ m−1
(hypothesis of induction on m, which is satisfied when m=2) and for generalized
(N −1)-means of order m with N ≥ m+1 (hypothesis of induction on N) and
show that it remains true for u above.
n
Let E ⊂ ΛN denote the co-null set on which (2.7) holds. There is a co-null set
S of Λ such that, for every x ∈ S, the subset E := {(x ,...,x ) ∈ ΛN−1 :
N xN 1 N−1
(ex1,...,xN−1,xN)∈E} is co-enull ineΛN−1. Let xNe∈ S be chosen once and for all.
If (x ,...,x )∈E , then by splitting the cases when i ≤N −1 and i = N
in th1e rightN-h−ae1nd sidxeeNof (2.6), we may rewrite (e2.7) weith xmN =xN as m
e
(2.8)
u (x ,...,x )+ u (x ,...,x ,x )→
n i1 im n i1 im−1 N
1≤i1<··X·<im≤N−1 1≤i1<···<Xim−1≤N−1
e
N
U(x ,...,x ,x ).
(cid:18) m (cid:19) 1 N−1 N
e
By Lemma 2.1, the set Ω := {(x ,...,x ) ∈ ΛN : (x ,...,x ) ∈ E for
1 N j1 jN−1 xN
every 1 ≤ j1 < ··· < jN−1e≤ N} is co-null in ΛN and, if (x1,...,xN) ∈ Ωe, then
(2.8) holds with (x ,...,x ) replaced with (x ,...,x ) and 1 ≤ j < ··· <
1 N−1 j1 jN−1 1 e
j ≤ N fixed. If so, the m-tuple (x ,...,x ) in the left-hand side of (2.8)
N−1 i1 im
becomes (x ,...,x ). Since j < ··· < j and i < ··· < i , it follows that
ji1 jim 1 N−1 1 m
j < ··· < j . We do not write down the corresponding formula (2.8) for this
i1 im
case,becausethereisasimpler(andotherwisecrucial)waytoformulateitwithout
involving sub-subscripts, as detailed below.
There are only N different ways to pick N −1 variables x ,...,x with in-
j1 jN−1
creasing indices among the N variables x ,...,x , that is, by omitting a different
1 N
variable x ,1 ≤ k ≤ N. For every such k, the variables x ,...,x are then
k j1 jN−1
x ,...,x ,...,x where, as is customary, x means that x is omitted. Thus, as
1 k N k k
i ,...,i run over all the indices such that 1 ≤ i < ··· < i ≤ N − 1, the
1 m 1 m
correspconding m-tuples (x ,...,x ) rucn over all the m-tuples (x ,...,x ) in
ji1 jim ℓ1 ℓm
whichthe variablex doesnotappear,thatis,overthem-tuples(x ,...,x )with
k ℓ1 ℓm
1≤ℓ <···<ℓ ≤N and ℓ 6=k,...,ℓ 6=k.
1 m 1 m
GENERALIZED MEANS 7
In light of the above, when (x ,...,x ) is replaced with (x ,...,x ) in (2.8)
i1 im ji1 jim
and the variable x does not appear, the formula (2.8) takes the form
k
(2.9) u (x ,...,x )+
n ℓ1 ℓm
X
1≤ℓ <···<ℓ ≤N
1 m
ℓ 6=k,...,ℓ 6=k
1 m
N
u (x ,...,x ,x )→ U(x ,..x ,...,x ,x ).
n ℓ1 ℓm−1 N (cid:18) m (cid:19) 1 k N N
X
1≤ℓ1 <···<ℓm−1 ≤N e N−1cvariables e
ℓ 6=k,...,ℓ 6=k | {z }
1 m−1
Recallthat(2.9)holdsforevery1≤k ≤N andevery(x ,...,x )∈Ω.Byaddingup
1 N
the relations (2.9) for k =1,...,N, we get (after replacing ℓ ,...,ℓ with i ,...,i )
1 me 1 m
N
(2.10) u (x ,...,x )+
n i1 im
Xk=1 1≤i <X···<i ≤N
1 m
i 6=k,...,i 6=k
1 m
N
N
u (x ,...,x ,x )→ l(x ,...,x ),
n i1 im−1 N (cid:18) m (cid:19) 1 N
Xk=1 1≤i1 <·X··<im−1 ≤N e e
i 6=k,...,i 6=k
1 m−1
for every (x ,...,x )∈Ω, where l(x ,...,x ):=ΣN U(x ,...x ,...,x ,x ), i.e.,
1 N 1 N k=1 1 k N N
e e c e
(2.11) l=NG (U(·,x )).
N−1,N N
e e
Let 1 ≤ i < ··· < i ≤ N be fixed. In the first double sum of (2.10),
1 m
u (x ,...,x ) appears exactly once for every index 1 ≤ k ≤ N such that k 6=
n i1 im
i ,...,k 6=i . Evidently, there are N −m such indices k and so
1 m
N
u (x ,...,x )=
n i1 im
kX=1 1≤i <X···<i ≤N
1 m
i 6=k,...,i 6=k
1 m
N
(N −m) u (x ,...,x )=(N −m) U (x ,...,x ).
n i1 im (cid:18) m (cid:19) n 1 N
1≤i1<X···<im≤N
8 IRINANAVROTSKAYAANDPATRICKJ.RABIER
The seconddouble sumin(2.10)has the same structureas the firstone,with m
replaced with m−1. Thus,
N
u (x ,...,x ,x )=
n i1 im−1 N
kX=1 1≤i <··X·<i ≤N
1 m−1 e
i 6=k,...,i 6=k
1 m−1
(N −m+1) u (x ,...,x ,x )=
n i1 im−1 N
1≤i1<··X·<im−1≤N
e
N
m G (u (·,x ))(x ,...,x ).
(cid:18) m (cid:19) m−1,N n N 1 N
e
Hence, (2.10) boils down to (recall (2.11))
(N −m)U +mG (u (·,x ))→NG (U(·,x )) on Ω.
n m−1,N n N N−1,N N
By (2.7), Un →U on E. Thus, e e e
(2.12) G (u (·,x ))→m−1(NG (U(·,x ))−(N −m)U) on Ω∩E.
m−1,N n N N−1,N N
Since Ω∩E is co-nullein ΛN, this means that the seequence G (u (·,xe )) has
m−1,N n N
ana.e. finite limit. Therefore,the hypothesis of induction on m ensures that there
is an ae.e. finite function v on Λm−1 such that u (·,x ) → v a.e. on Λm−e1. Thus,
n N
G (u(·,x )) → G (v) on some co-null subset T of ΛN−1 and
m−1,N−1 N m−1,N−1 N−1
then, from (2.8), e
e
N m
(2.13) G (u )→ U(·,x )− G (v) on T ∩E .
m,N−1 n N −m N N −m m−1,N−1 N−1 xN
e
Since T ∩ E is co-null in ΛNe−1, this shows that G (u ) has an a.e.
N−1 xN m,N−1 n
finite limit. Thaet un has an a.e. finite limit u thus follows from the hypothesis of
induction on N and so, by the first part of the proof, U =G (u). (cid:3)
m,N
The following corollary gives a short rigorous proof of an otherwise intuitively
clear property.
Corollary 2.4. A generalized N-mean U of order m has a unique kernel u (up to
modifications on a null set).
Proof. If u and v are two kernels of U, set u = u if n is odd and u = v if n is
n n
even. Then, G (u ) = U a.e. for every n. By Theorem 2.3, u converges a.e.,
m,N n n
which implies u=v a.e. (cid:3)
Significant differences between m =1 and m>1 are worth pointing out (with-
out proof, for brevity). When m = 1, a stronger form of Lemma 2.2 holds: If
G (u )→U a.e. with U finite on a subset of ΛN of positive dNx measure, then
1,N n
u → u a.e. and u is finite on a subset of Λ of positive dx measure (the converse
n
is clearly false). There is no such improvement of Theorem 2.3 if m > 1. It is not
hard to find examples (variants of Example 2.1) when G (u ) has an a.e. limit
m,N n
which is finite on a subset of ΛN of positive dNx measure, but u has no limit at
n
every point of a subset of Λ of positive dx measure.
GENERALIZED MEANS 9
3. Kernel recovery and measurability
As pointed out in the Introduction, there are rather simple ways to recover u
from G (u), but the resulting formulas are inadequate to answer even the most
m,N
basic measure-theoretic questions. Below, we describe a recovery procedure that
preserves all the relevant measure-theoretic information.
ThefactthatthekernelofageneralizedN-meanofordermisunique(Corollary
2.4)meansthatG hasaninverseK (“kerneloperator”),i.e.,K (U)=u.
m,N m,N m,N
Bythe linearityofG , itfollowsthat K (defined onthe space ofgeneralized
m,N m,N
N-means of order m) is also linear.
Explicit formulas for K and K will be given but, when m ≥ 2 and N ≥
1,N m,m
m+1, K will only be defined inductively, in terms of K and K
m,N m−1,N m,N−1
(andalsoG andG ). Thismakesitpossible,bytransfiniteinduction
m−1,N−1 N−1,N
(as in the proof of Theorem 2.3) to recover K for arbitrary m and N ≥m.
m,N
First, G = K = I is obvious. To define K , we return to the proof of
m,m m,m 1,N
Lemma 2.2 in the case when the sequences u and U are constant and therefore
n n
equal to their a.e. limits u and U, respectively. If so, (2.5) yields the kernel of U :
(3.1) K (U)=
1,N
NU(·,x ,...,x )+U(x ,...,x )−NG (U(·,x ,...,x ))(x ,...,x ),
2 N 1 N 1,N 2 N 1 N
where x2,...,xNeis arbietrarily chosen in some suitable co-enull suebset of ΛN−1 and
(x ,...,x ) is arbitrarily chosen in some suitable co-null subset of ΛN. (Of course,
1 N
these ceo-nullesubsets depend on U.)
Notethat(3.1)immediatelyshowsthatK (U)ismeasurableifU ismeasurable
1,N
sinceitisalwayspossibletochoosex ,...,x suchthatU(·,x ,...,x )ismeasurable
2 N 2 N
(the extratermU(x ,...,x )−NG (U(·,x ,...,x ))(x ,...,x )in(3.1)is justa
1 N 1,N 2 N 1 N
constant). e e e e
To define K in terms of K andeK e when m≥2 and N ≥m+1,
m,N m−1,N m,N−1
we return to the proofof Theorem 2.3 when the sequences u and U are constant
n n
and therefore equal to their a.e. limits u and U, respectively. If so, u = u and
n
(2.12) is the a.e. equality
G (u(·,x ))=m−1(NG (U(·,x ))−(N −m)U),
m−1,N N N−1,N N
wherex isarbitrarilychoseninsomesuitableco-nullsubsetofΛ.Thisshowsthat
N e e
NG (U(·,x ))−(N −m)U is a generalizedN-mean of order m−1 and that
N−1,N N
e
(3.2) u(·,x )=m−1K (NG (U(·,x ))−(N −m)U).
eN m−1,N N−1,N N
Next, (2.13) is the a.e. equality (since v =u(·,x ) in (2.13))
e N e
N m
G (u)= U(·,x )− e G (u(·,x )).
m,N−1 N m−1,N−1 N
N −m N −m
This shows that NU(·,x )−mG e (u(·,x )) is a generalizeed (N −1)-mean
N m−1,N−1 N
of order m and, by (3.2), that K (U)=u is given by (when 2≤m≤N −1)
m,N
e e
(3.3) (N −m)K (U)=
m,N
K (NU(·,x )−G (K (NG (U(·,x ))−(N −m)U))).
m,N−1 N m−1,N−1 m−1,N N−1,N N
If an a.e. defiened function on Λk is saidto be symmetriceif it coincides a.e. with
aneverywheredefinedsymmetricfunction,thenageneralizedN-meanissymmetric
ifandonlyifitskernelissymmetric. Indeed,itisobviousthatthe operatorsG
m,N
10 IRINANAVROTSKAYAANDPATRICKJ.RABIER
preserve symmetry. Conversely, since K = I and K (obviously) preserve
m,m 1,N
symmetry, K preserves symmetry by (3.3) and transfinite induction.
m,N
Theorem 3.1. A generalized N-mean U of order m is measurable if and only if
its kernel u is measurable.
Proof. ItisobviousthatthemeasurabilityofuimpliesthemeasurabilityofU.From
now on, we assume that U is measurable and prove that u is measurable.
This is trivial when m = N and was already noted after (3.1) when m = 1. If
m≥2 andN ≥m+1, we proceed by transfinite induction, thereby assuming that
the theorem is true if m is replaced with m−1 and, with m being held fixed, if N
is replaced with N −1.
In (3.3), choose x so that U(·,x ) is measurable. This is possible since x is
N N N
arbitrary in a co-null subset of Λ. In particular, G (U(·,x )) is measurable
N−1,N N
(recall that the meaesurability of theekernel always implies the measurability oef the
generalized mean) and so NG (U(·,x ))−(N −m)U isemeasurable. Since
N−1,N N
this is a generalizedN-meanoforderm−1 (see the proofof (3.3)), the hypothesis
ofinductiononmensuresthatitskernelisemeasurable. ItfollowsthatNU(·,x )−
N
G (K (NG (U(·,x ))−(N −m)U)) is measurable. Since this
m−1,N−1 m−1,N N−1,N N
isageneralized(N−1)-meanofordermwithkernel(N−m)K (U)=(N−em)u
m,N
(see once again the proof of (3.3)), iet follows from the hypothesis of induction on
N that (N −m)u is measurable. Thus, u is measurable. (cid:3)
To complement the measurability discussion, we investigate essential bounded-
ness. In the next theorem, ||·||Λk,∞ denotes the L∞ norm on Λk.
Theorem 3.2. (i) If u is a measurable function on Λ, its generalized N-mean
U :=G (u) is essentially bounded above (below) if and only if u is bounded above
1,N
(below). Furthermore, esssupU =esssupu and essinfU =essinfu. In particular,
u∈L∞(Λ;dx) if and only if U ∈L∞(ΛN;dNx) and, if so, ||U||ΛN,∞ =||u||Λ,∞.
(ii)Let1≤m≤N beintegers. Ifu∈L∞(Λm;dmx),thenG (u)∈L∞(ΛN;dNx)
m,N
and ||Gm,N(u)||ΛN,∞ ≤ ||u||Λm,∞. Conversely, if U ∈ L∞(ΛN;dNx) is a general-
ized N -mean of order m, then K (U) ∈ L∞(Λm;dmx) and there is a constant
m,N
C(m,N) independent of U such that ||Km,N(U)||Λm,∞ ≤C(m,N)||U||ΛN,∞.
Proof. (i)Ifa∈Randu≥aonasubsetE ofΛofpositivedxmeasure,thenU ≥a
on EN, of positive dNx measure. This shows that esssupU ≥ esssupu. On the
otherhand,itisobviousthatU ≤esssupua.e. onΛN,sothatesssupU ≤esssupu.
This shows that esssupU =esssupu. Likewise, essinfU =essinfu.
(ii) The first part (when u ∈ L∞(Λm;dmx)) is straightforward. Suppose now
that U ∈ L∞(ΛN;dNx) is a generalized N-mean of order m. If m = 1, the result
withC(1,N)=1followsfrom(i)anditistrivialwhenm=N (withC(m,m)=1).
To provethe existence of C(m,N) in general,we proceedonce againby transfinite
induction. The hypothesis of induction is that m ≥ 2,N ≥ m+1 and that the
constants C(m−1,N) and C(m,N −1) exist.
In(3.3),choosexN suchthatU(·,xN)ismeasurableandthat||U(·,xN)||ΛN−1,∞ ≤
||U||ΛN,∞ (which holds for a.e. xN ∈ Λ). Then, ||GN−1,N(U(·,xN)||ΛN,∞ ≤
||U||ΛN,∞ and it roeutinely follows freom (3.3) that C(m,N) may be deefined by
e e
C(m,N −1)
C(m,N):= (N +(2N −m)C(m−1,N)).
N −m
(cid:3)
