The Annals
ofProbability
The
Annals 0/
Probability
1987,
No.2,
804-813
1987, Vol.
Vol. 15,
15, No.
2, 804-813
MEASURES'l
PROBABILITY MEASURES
GENERAL PROBABILITY
PARTITIONING GENERAL
PARTITIONING
By
THEODOREP.
P. HILL
HILL
BY THEODORE
Georgia
Institute of
ofTechnology
Technology
GeorgiaInstitute
measurable
on the
the same
same measurable
measureson
are probability
Suppose ILl'l,...,
Suppose
.•. ' ILn
probability measures
An are
thereis
is aa .
Y). Then
mass aa or
or less,
less,there
of each
each ILi
havemass
Thenif
if all
all atoms
atomsof
space (0,
(2, !F).
space
Ai have
2 ~(a)
of 02 so
so that
that ILi(A
forall
all ii ==
measurablepartition
AI' ... ' An
An of
measurable
partition Al,...,
pxi(Ai)
Vn(a) for
i) ~
1,...
linearnonincreasing
nonincreasing
givenpiecewise
is an
an explicitly
explicitly
...,,n,
n, where
where~(.)
1,
given
piecewise linear
Vn(.) is
forall
all nn
is attained
attainedfor
thebound
bound~(a)
on [0,1].
[0,1].Moreover,
Moreover,
continuous
function
continuous
function
on
the
Vn(a)is
a. Applications
theory,
and all
all a.
are given
to Ll
statistical
decisiontheory,
and
are
L I spaces,
decision
spaces,to
to statistical
givento
Applications
theclassical
and
case.
and to
to the
nonatomic
case.
classicalnonatomic
meaany nonatomic
nonatomicprobability
space of
of any
1. Introduction.
The underlying
1.
underlying space
probability meaIntroduction. The
havingmeasure
measure
subsetseach
each having
sure
be partitioned
partitioned into
inton
n measurable
measurablesubsets
suremay
alwaysbe
mayalways
if there
on
measureson
nonatomicprobability
exactly
if
probability measures
are kk nonatomic
More generally,
thereare
1/n. More
generally,
exactlylin.
space
ofthe
thespace
the
Neyman [6]
partition of
measurable
partition
thereis
is aa measurable
[6] showed
showedthere
space,Neyman
thesame
samespace,
each
1/n to
to each
measureexactly
exactlylin
into
probability assigns
assignsmeasure
into n
subsetsso
so that
that each
each probability
n subsets
of n
n
subset,
Nile" [4].
In the
the case
case of
[4]. In
"Problemof
of the
the Nile"
solvingFisher's
Fisher's"Problem
therebysolving
subset,thereby
Knaster[7]
[7] gave
gave aa
continuous
probability measures,
Banach and
and Knaster
Steinhaus,Banach
measures,Steinhaus,
continuousprobability
that
theproperty
property
n sets
setswith
withthe
that
inton
practical method
aa partition
partition into
determining
methodfor
fordetennining
practical
results,
the ith
of these
theseresults,
ith measure
of the
the ith
ith subset
subsetis
is at
at least
1/n. Extensions
Extensionsof
keastlin.
the
measureof
many
theorem
[5]
ofevery
everynonatomic
nonatomic
("the range
rangeof
theorem
[5] ("the
convexity
usingLyapounov's
Lyapounov'sconvexity
manyusing
and
compact)")and
is convex
convex(and
(and compact)")
finite-dimensional,
vector
measureis
valued(finite)
(finite)measure
vectorvalued
finite-dimensional,
Dubins
generalizations
were
by Dvoretzky,
Wald
[2]
[2] and
and Dubins
and Wolfowitz
Wolfowitz
Dvoretzky,
Waldand
wereobtained
obtainedby
generalizations
and
and Spanier
[1].
Spanier[1].
In
results
haveatoms,
atoms,
In general,
ofthe
failif
ifthe
themeasures
measureshave
resultsfail
all of
theabove-mentioned
above-mentioned
general,all
and
purpose of
paper to
some
best possible
possible partitioning
partitioning
thispaper
somebest
to detennine
determine
and it
it is
is the
the purpose
ofthis
size
bounds as
of
size of
ofthe
theatoms.
atoms.
bounds
as aa function
function
ofthe
themaximum
maximum
adThroughout
paper (0,
but any
measurablespace
space adthis paper
any measurable
(R,Borels),
Borels),but
F) == (IR,
Throughoutthis
(Q, $i)
choiceis
is mainly
mitting nonatomic
probability measures
particular choice
willdo;
measureswill
thisparticular
mainly
nonatomicprobability
do; this
mitting
is nonatomic
ifand
and
on (IR,
for
since
if
measureJLit on
nonatomic
convenience
sinceaa measure
(R,Borels)
Borels)is
fornotational
notationalconvenience
E
0
ifJL({x})
=
for
all
x
R.
only
=
for
all
x
E
IR.
only if
tt({x})
°
a Ee [0,1],
DEFINITION. For
DEFINITION.
For each
each a
[0,1],
fJJ(
a) == {JL:
probability measure
measureon
is aa probability
on (0,
F)
(62,$i)
9 (a)
(,i: JL
,uis
E 02}.
< aa for
withJL({x})
X EO}.
forall
all x
with
(4{x}) ~
DEFINITION. V
DEFINITION.
[0,1]
Vn:
n : [0,1]
Figure 1)
satisfying
Figure
1) satisfying
(1)
(1)
-+
--+
1
function(see
the unique
] is
[0,
n 1]
is the
nonincreasingfunction
unique nonincreasing
[0, n-
Vn(a)
Vn(a)
=
= 1
1- -
k(n
1)a
k(n -- l)a
Received April
1985.
Received
April1985.
Research partially
partially supported
by NSF
NSF Grant
DMS-84-01604.
GrantDMS-84-01604.
'Research
by
supported
62C20.
AMS 1980
subject classifications.
Primary
6OAI0,
6OE15,
1980subject
60E15,62C20.
secondary
AMS
Primary
60A10,28A99;
28A99;secondary
classifications.
cake-cutKey words
phrases. Optimal-partitioning
inequalities,
atomic
probability measures,
cake-cutatomicprobability
measures,
wordsand
inequalities,
Optimal-partitioning
andphrases.
Key
rules.
ting, fair
problems, minimax
decision
minimax
decisionrules.
fairdivision
divisionproblems,
ting,
I
804
PARTITIONINGGENERAL
GENERAL PROBABILITY
PARTITIONING
PROBABILITY MEASURES
MEASURES
805
805
V,(a)
2i
1
n=2
3 II
I3N_
\~~~~~
I
I
I
n=3
I
I
I
I
I
~~~~~~~1
1\
~---------------~----------~---~a
1
1
22
FIG. 1.
1.
FIG.
Graphs
ofl'2
andV3.
V2and~.
Graphs
of
for
for
a E [(k + 1)k-1((k + 1)n - 1) 1, (kn - 1)-1],
~ 1.
for
forall
all kk 2
1.
The main
The
two
mainresults
resultsof
ofthis
thispaper
are the
thefollowing
twoclosely
relatedtheorems.
theorems.
paperare
following
closelyrelated
> 11 there
THEOREM
Let JL,. Ee 9(a).
fJJ(a). Then
for each
THEOREM 1.1.
1.1. Let
Thenfor
each nn >
thereexists
existsaa measurable
measurable
partition {Ai}i=l
partition
{Ai} n1 of
of gQ satisfying
satisfying
(2)
JL(Ai)~Vn(a),
,u(Ai) 2 V.(a),
foralli=l,
... ,n;
all i 1.,n;
for
moreover,
V
possible bound
for aU
a.
is the
thebest
bestpossible
boundin
in (2),
and is
is attained
attainedfor
all a.
moreover,
(2), and
V.n is
THEOREM 1.2.
1.2. Let
Then there
thereexists
exists a
a measurable
THEOREM
Let JLl"'"
JLn Ee gJ(
a). Then
partimeasurableparti,ui,..., it.
9(a).
tion
1 of
satisfying
.
tion{Ai}i==
of 0
S satisfying
{Ai},n1
(3)
(3)
JLi(A i ) ~
2 Vn(a),
,Ai(Ai)
V,(a),
all i == 1,
1,...,
foralli
... , n;
for
n;
a.
again,
possible bound
for all
and is
is attained
attainedfor
all a.
is the
thebest
bestpossible
boundin
in (3),
again, V
(3), and
V.
n is
REMARK.
bound
in the
sense: the
REMARK. Theorems
Theorems1.1
1.1 and
1.2 are
are "dual"
"dual" in
the following
the bound
and 1.2
followingsense:
= ...
...
in Theorem
1.1 follows
of Theorem
Theorem1.2
1.2 by
(2)
by taking
=
JLn'
(2) in
Theorem1.1
followsfrom
from(3)
(3) of
takingJLl
,u =
= An,
wnereas the
sharpness of
from
in Theorem
1.2 follows
from
whereas
the sharpness
of the
Theorem1.2
followssimilarly
the bound
bound(3)
(3) in
similarly
in Theorem
the
1.1.
of(2)
the sharpness
Theorem1.1.
sharpnessof
(2) in
A
interpretation
of
based on
by
A "cake-cutting"
of Theorem
1.2 based
Theorem1.2
on aa description
"cake-cutting"
interpretation
description
by
and
Spanier
[1]
is
this.
Suppose
a
cake
0
is
to
be
divided
among
Dubins
is
to
be
divided
n
Dubins and Spanier[1] is this.Suppose a cake 2
amongn
806
T.
T. P.
P. HILL
HILL
different
people
whosevalues
values {JLi}i-1
of different
portionsof
of the
the cake
cake may
may differ
differ
[here
people whose
portions
[here
{fliij1 of
A
of
to
i].
JLi(A)
represents
the
value
of
piece
A
to
person
i].
Then
if
no
one
values
any
the
value
represents
piece
person
Then
if
no
one
values
any
1ii(A)
crumb
of the
the cake)
morethan
than a,
crumb(indivisible
(indivisibleportion
portionof
cake) more
a, the
the cake
cake may
may divided
dividedso
so
that each
person receives
piece he
values
and
in
general
he himself
at least
each person
receivesaa piece
himself
valuesat
leastV:(a),
in
that
and
general
V.(a),
it
possible to
it is
is not
not possible
to do
do better.
better.
EXAMPLE
1.3.
EXAMPLE
1.3. Suppose
Supposethree
threepeople
peoplemust
mustdivide
divideaa cake,
cake,and
and each
eachagrees
agreesthat
that
3
no crumb
the
no
is aa
crumbis
is worth
worthmore
morethan
than10thevalue
valueof
ofthe
thewhole
wholecake.
cake.Then
Thenthere
thereis
10-3
way of
piece, in
in such
ofcutting
thecake
cakeinto
intothree
eachperson
such
way
cuttingthe
threepieces,
pieces,and
and giving
givingeach
personaa piece,
aa way
person values
thateach
each person
hisown
valueshis
ownpiece
at least
leastV:(a)
waythat
pieceat
V3(10-3) == 83/250
83/250
V.(a) == ~(10-3)
and
possible to
and in
in general
generalit
it is
is not
notpossible
to do
do better.
better.
(A
of
(A similar
similarinterpretation
interpretation
of Theorem
Theorem1.1
1.1 is
is also
also possible.
possible.Suppose
Supposeaa cake
cake of
of
total
be cut
piece
is to
to be
volume(or
oneis
cutinto
inton
n pieces
so that
thatthe
thesmallest
smallestpiece
totalvolume
(or weight)
weight)one
piecesso
has
has as
as large
If each
largeaa volume
volumeas
as possible.
possible.If
each atom
atom(or
(or molecule,
molecule,or
or crumb,
crumb,or
or other
other
indivisible
piece) has
partitioning the
in an
the
indivisiblepiece)
has volume
volumeaa or
or less,
less, then
then in
an optimal
optimalpartitioning
in
(
smallest
piece has
a),
and
in
general
this
is
the
best
possible
this
the
best
smallestpiece
has volume
volumeat
at least
leastV
and
is
general
possible
V.(a),
n
bound.)
bound.)
limitof
Intuitively,
it
case
Intuitively,
it is
is clear
clearthat
thatthe
thenonatomic
nonatomic
case is
is the
thelimit
ofthe
thegeneral
generalcase
case
as
themaximum
maximumatom
atomsize
size approaches
zero.
as the
approacheszero.
on
nonatomic
measureson
COROLLARY
measures
COROLLARY1.4
1.4 ([1],
are nonatomic
([1], [2],
[2],[7]).
[7]). Suppose
SupposeJL1' •••
t.., JL n are
Q
so
that
Thenthere
existsaa measurable
thereexists
measurable
(g,
partition {A
}i-1
of
so
that
partition
of
(Q, ~).
_>). Then
{Ai}in~1
i
°
1,
JLi(A i ) ~ nn-1,
Ai(Ai)
foralli
... , n.
for
alli == 1,
1,...,
n.
This
paper is
proof of
This paper
is organized
organizedas
as follows:
follows:Section
Section22 contains
containsthe
the proof
of Theorem
Theorem
1.1;
proof of
observations
about
Theorem1.2;
further
observations
aboutthe
the
theproof
ofTheorem
Section44 further
1.1; Section
Section33 the
1.2;Section
upper
bound function
Vn(a);
and Section
severalapplications
to LL11
function
containsseveral
upperbound
Section55 contains
applicationsto
V.(a); and
function
spaces
function
and statistical
statisticaldecision
decisiontheory.
spacesand
theory.
The main
mainobjective
ofthis
this
2.
probability measure.
2. Partitioning
measure. The
Partitioningaa single
single probability
objectiveof
will denote
denotethe
the
this paper,
section
prove Theorem
this
Theorem1.1.
1.1. Throughout
II will
sectionis
is to
to prove
Throughout
paper,I1!9
is aa sub-a-algebra
of~,
collection
partitions
of
of
where~
and
of 0,2,where
of ~-measurable
9-measurable
collectionof
partitions
C is
sub-a-algebra
Y, and
a(CC)
W.
the a-algebra
will denote
denote the
a-algebragenerated
generatedby
by ~.
a(W ) will
DEFINITION
DEFINITION
on (0,
Then
2.1.
JL is
measure
is aa probability
measureon
2.1. Suppose
Suppose[t
probability
(E2,$).
F). Then
}
e II.F}
min {JL(A
Un(JL)
sup ~n
i )}: {A
{pI(Ai)}:
{Ai}i }7-1 EHE
Un(lt) = sup{
l~t~n
and
and
Un(a) = inf{ Un():A) E 9(a)}
:.LEMMA
Fix aa EE (0,1].
For each
JL EE P9(a)
f!J(a) there
purely atomic
thereexists
existsaa purely
atomic
each At
2.2. Fix
SLEMMA 2.2.
(0,1]. For
1
flAEE9(a)
E f!J(a) having
at
most
2aatoms,
and
satisfying
and
havingat most2a-1 atoms,
satisfying
(4)
(4)
Un(fl)
Un(JL) ,
U(j) ~
<n(),
for
allnn ~
1.
for all
2 1.
PARTITIONING
PARTITIONINGGENERAL
GENERALPROBABILITY
PROBABILITY MEASURES
MEASURES
807
807
PROOF. The
The idea
idea of
of the
the proof
PROOF.
is simply
proofis
simplythat
that collapsing
collapsingmass
mass to
to atoms
atoms
reducesthe
thepartitioning
partitioning
optionsavailable,
andthus
reduces
options
available,and
thusreduces
reducesUn;
Un;for
forcompleteness
completeness
thefirst
willbe
firststep
be given
insome
stepwill
givenin
somedetail.
A == {Xl'
detail.Let
Let A
the
}} Cc 0Q denote
x2,...
denotethe
the
{x1,X
2,
atoms of
of J.LAtand
and AC
If J.L(AC)
Ac=Q\A.
let AI'
atoms
= 0 \A. If
> 0, let
,
be
a
measurable
tL(AC)>0,
A1,A
A2,
...
be
a
measurable
2
of AC
Ac satisfying
0 << j(Ai)
partitionof
< aa for
satisfying
forall
all i,i, which
partition
J.L(A i ) ~
whichis
is possible
possiblesince
sinceJ.L,uis
is
nonatomic
on A
Ac.
Foreach
fixYi
eachi,i,fix
andlet
nonatomic
on
letJ.Ll
C. For
a) be
69(a)
be the
thepurely
purelyatomic
atomic
yiEE Ai'
Ai,and
AlEE f!J(
probability
measuredefined
defined
byJ.Ll({Xi})
and J.Ll({Yi})
probability
measure
by
SinceJ.LA
lt({xi}) == J.L({x
t({xi})
(Ai).
Al({Yi})== J.L(A
i }) and
i ). Since
f
restrictedto
to o({x
A2,...)
{x2},
is
restricted
...
)
is
isomorphic
to
fl
restricted
to
},
A
,
isomorphic
to
restricted
to
a({xllJ ,) AI'
Al, {x
2
2
....
and
since
...
are
},
A
,
.
o(
{Xl}' {Yl}'
.),
and
since
(recall
{Xl}'
AI'
{x
..
are
disjoint)
},
{Y2}'
·
·
),
{Y1},{x
{x2},
{y2}
a({Xl},
(recall
{x2},
{xl},
A2,
2
Al,
disjoint)
2
2
{x2},
A2,
...) ) Cc $i,
it follows
follows
A, it
thatUn(J.Ll)
U,(tt)for
forall
all nn ~2 1.
o(
{Xl}' AI'
that
1.
a({xl},
Al,{x
Un(ttl)~< Un(J.L)
2}, A
2, ...
The next
nextstep
is to
to replace
stepis
replaceJ.Ll
by aa purely
The
purelyatomic
atomicmeasure
measurewith
witheach
each atom
atom
A1 by
havingmass
massat
at least
least a2a2-1l (and
having
(andhence
hencehaving
havingat
at most
most2a2a-1l atoms).
atoms).This
Thisis
is done
done
firstcombining
by first
tail {x
thetail
combining
{XN},
{YN}, {XN+l}'
... into
{XN+1), {YN+l}'
intoone
{YN+1) ...
oneatom
atom(where
(where
by
the
N}, {YN}'
+ J.Ll({Yi})]
to reduce
reduce to
to aa finite
finitenumber
numberof
of atoms,
< a)
Ei-N[J.Ll({Xi})
~
and then
then
EL=N[tNl({Xi}) +
a) to
iy({yij)]
atoms, and
< a/2.
by
any
twoatoms
withmass
atomswith
by repeatedly
repeatedlycombining
combining
mass ~
anytwo
a/2. 00
°
LEMMA 2.3.
2.3. For
For each
each aa EE [0,1]
and nn ~
[0,1] and
LEMMA
2 1,
1, there
thereexists
existsaa J.L,AEE f!J(a)
69(a) and
and aa
E
partition
satisfying
partiti0':t {Ai}i-l
{Ai}i1 l E II,.
HEwsatisfying
(5)
(5)
Un(a) = #(Al) <
(A2) <
...
<
(A
°
= 0 (which
PROOF. For
For aa =
willnot
(whichwill
notbe
be needed
neededin
in this
PROOF.
thispaper)
paper)the
the result
resultis
is an
an
easy consequence
consequence (taking
easy
J.Ll =
= ...
= J.Ln)
of
Lyapounov's
convexity
theorem
[5].
(takingAl
of
=
Lyapounov's
convexity
theorem
...
un)
l }, and
> max{n,
Fix
(0,1] and
and kk >
Fix aa E
2a-1},
E (0,1]
max.{n,2apoints xl,...,
Xl, ... ,Xk
and choose
choosekk distinct
distinctpoints
Xk
=
in
U. By
E
and
Lemma
2.2,
in O.
By the
the definition
definition of
of Un(a)
and
Lemma
2.2,
Un(a)
=
inf{Un(J.L):
J.L
E
f!J(a,
k)},
=9(a,
k)},
Un(a)
Un(a) inf{Un(A):At
where
69(a, k)
k) =
ElkA1({xi1) == 1).
Since f!J(a,
where f!J(a,
= {AE
{J.L E-9(a):
f!J(a): Ef-lJ.L({Xi})
I}. Since
k) is
is compact,
compact, and
and
9(a, k)
since
since Un
is aa continuous
continuous function
function of
of ,A
J.L Ec
E 6(a,
f!J( a, k),
k), inf{Un(t):
inf {Un(J.L): ,A
J.L EE 9(a,
f!J( a, k)}
k)} is
is
Un is
attained
attained by
by some
some fta
fl E c(a,
f!J( a, k).
k). Since
Since the
the support
support of
of ft
fl is
is aa finite
finite,.. set
set (subset
(subset of
of
{X1
{Xl' ...,
... ' Xkj),
Xk})' it
it is
is clear
clear that
that there
there is
is aa partition
partition {Ai},n'1
{Ai}i=l EcI
E H
II,. satisfying
satisfying (5)
(5) with
with
ft
fl in
in place
place of
of ,A.
J.L. 0
0
PROOF
PROOF OF
OF THEOREM
THEOREM 1.1.
1.1. Fix
Fix n
n >> 1
1 and
and k
k 2
~ 1
1 and
and let
let aa E
E I(n,
I(n, k),
k), where
where
l «k +
I(n, k)
k) =
= [(k
[(k +
+ 1)k
l)k+ 1)n
l)n -- 1)-t,(kn
1)-1] cC (0,1).
I(n,
-((k
1)- 1,(kn -- 1)-1]
It
It first
first will
will be
be shown
shown that
that on
on I(n,
I(n, k),
k), Vn
Vn =
= Un.
Un. By
By Lemmas
Lemmas 2.3
2.3 and
and 2.2
2.2 there
there
exists
fJJ(a) with
with at
at most
most 2a-1
2a- l atoms,
and aa
atoms, and
exists aa purely
purely atomic
atomic measure
measure lJ.L cE 9P(a)
partition
partition {Ai}i.
{Ai}i-l1 E
E HI
II,.
satisfying (5).
(5).
_ satisfying
Suppose, by
by way
way of
of contradiction,
contradiction, that
that tt(Al)
J.L(A l ) << 1
1 -- k(n
k(n -- 1)a.
l)a. Since ,u
J.L is a
Suppose,
probability
J.L(Ui-2Ai)
k(n -- 1)a,
l)a, and
and since
since the
the {Ai}
{Ai} are
are disjoint,
disjoint,
probability measure,
measure, tt(UW
2Ai) >> k(n
this
this implies
implies that
that for
for some
some jj EE .{2,
.{2, 3,
3, ..
....,, n},
n}, tt(Aj)
J.L(Aj ) >> ka.
ka. Since
Since ,A
J.L is
is purely
purely atomic
atomic
and
and in
in 9(a),
f!J(a), Aj
A j must
must contain
contain at
at least
least k
k+
+ 11 D-atoms.
J.L-atoms. Let
Let {x;}
{Xj} E
E Aj
A j be
be the
the
smallest
smallest atom
atom in
in Aj
A j (which
(which exists
exists since
since At
J.L has
has only
only aa finite
finite number
number of
of atoms)
atoms) and
and
observe
observe that
that
(6)
(6)
ju(Al U {xj}) > #(Al).
T.
T. P.
P. HILL
HILL
808
Since
in Aj,
A j , and
k + 11 atoms
in
atomin
and there
thereare
atomsin
is the
smallestatom
are at
least k
thesmallest
at least
Since {Xj}
{x;} is
this
implies
A
j'
this
implies
Aj,
(7)
(7)
2(k + 1)-la >~ 1- k(n - l)a,
IL(A
(A j ) >
+ 1)-l
> kk2(k
+ 1) la
1 - k(n - 1)a,
k(k +
1)f1lL(Aj)
j \ {Xj})
Au(Aj\
{xj}) 2~ k(k
1
where
~ (k
[(k +
in (7)
+ l)k+ l)n
last inequality
followssince
since aa 2
1)n -- 1]-1.
1]1.
(7) follows
(k +
1)k-1[(k
where the
the last
inequalityin
>
If
(6)
the
thentogether
contradict
theassumed
assumedoptimality
optimality
together
(6) and
and (7)
(7) contradict
If JL(A
,.(A2)
,t(Al),
1 ), then
2 ) > IL(A
the
= IL(A
ifIL(A
repeatthe
of ILAtand
and the
the partition
otherwise
[i.e.,if
(5) of
partition {A
[i.e.,
(5)
{Ai}ff=1;
,A(A2)
,A(Al)],
1 )], repeat
i }?==l; otherwise
2) =
in the
the
procedure with
A 2 , etc.
finitenumber
of sets
sets in
with A2,
thereare
are only
numberof
etc. Since
Since there
onlyaa finite
procedure
thatUn(lL)
partition, eventually
such
is
suchaa contradiction
is reached.
reached.This
This implies
contradiction
impliesthat
2
partition,
eventually
Un(tt)~
> V
thatUn
on I(n,
V
I( n, k).
and hence
hencethat
k).
Vn
Un~
Vn(a),
n on
n( a), and
<V
withkn
kn -- 11
To
be aa purely
measurewith
let fla EE f!J(
9(a)a) be
purelyatomic
atomicmeasure
To show
showU
Vn(a),
Un(a)
n ( a), let
n ( a) ~
atoms
of
mass
a,
and
one
atom
of
mass
1
a(kn
1).
[Since
a
E
I(n,
it
k), it
atoms of mass a, and one atom of mass 1 - a(kn - 1). [Since a I(n, k),
< a.]
< 11 -- a(kn
an optimal
forflf has
has
follows
partition for
that00 ~
followsthat
1) ~
a.] Clearly
Clearlyan
optimalpartition
a(kn -- 1)
(k -- l)a
fl(A
= (k
+ 11-- a(kn
1)a ~
a(kn- - 1)
-(A 1 ) )=
1)
< ka
=
1 -- k(n
ka
= 1
k(n -- l)a
1)a ~
=
=
... = fl(A
C fl(A
t(A2) 2 ) =
A nn)),
in fact
is attained
and in
thatVn(a)
attained(by
which
on I(n,
factthat
showsthat
that Un
(by fl).
a).
I(n, k),
k), and
whichshows
Vn(a) is
Un== VVn
n on
of
the
left
To
proof, observe
at
the
left
endpoint
of
value of
of V
at
observethat
thatthe
the value
endpoint
To complete
completethe
the proof,
Vn
n
+
k
that
of
I(n,
at
the
right
endpoint
of
I(n,
k
+
1),
that
as the
of V
at
the
is the
sameas
thevalue
valueof
I(n,
1),
right
endpoint
I(n, k)
k) is
the same
Vn
n
is,
1)(n - 1)y
l)y for
l)k- 1 [(k +
l)n - 1]-1
= (k
+1)k-1[(k
+ 1)nfor xx =
= 11 -- (k
+ 1)(n(k +
is, 11 -- k(n
k(n -- l)x
1)x =
(k +
1]-1
= «k
and
y =
was
defined
to
be
nonincreasing,
it
+ l)n
Then since
sinceV
was
defined
to
be
it
and y
nonincreasing,
((k +
1)n -- 1)-1.
1)-1. Then
Vn
n
mustbe
constanton
on [0,
U*=I(n,
k).
1]
\
must
be constant
1]
\
Uk==lI(n,
k).
[0,
= n= 00 are
is easy.
That
n11 and
and VVn(l)
are also
also attained
attainedis
easy.00
That Vn(O)
Vn(O)=
n(l) =
mainobjective
ofthis
this
3.
The main
objectiveof
3. Partitioning
several probability
probabilitymeasures.
measures. The
Partitioningseveral
section
prove Theorem
3.2 and
and Proposition
is to
Theorem1.2;
thefirst
firsttwo
tworesults
results(Lemma
Proposition
1.2;the
(Lemma3.2
sectionis
to prove
in nature.
3.3)
matrices and
purely combinatorial
in
combinatorial
nature.
and are
are purely
concernstochastic
stochasticmatrices
3.3) concern
is used:
used:
notation
thefollowing
following
notationis
Throughout
this
Throughout
thissection,
section,the
is the
theset
set of
of nn X
stochastic
~,kk is
matrices;
x kk stochastic
matrices;
Ykn
is the
set {I,
and
II-Ik
k is
of
of
... , k};
the collection
ofpartitions
ofthe
theset
{1,2,
2,...,
k}; and
collection
partitions
P
is
the
set
of
permutations
of
{I,
2,
...
,
k}.
of
is
the
set
of
k}.
{1,2,...,
permutations
Pk
k
DEFINITION
DEFINITION
Then
3.1. Suppose
3.1.
A == «(li,
j) EE Swn,
~,k.
k. Then
Suppose A
(0i, .)
max{ ~n {L ai,j}:)i) {~} 7==1 Ilk}.
W~(A) == max kmin{
W:(A)
l~l~n
3.2. For
3.2.
For each
A == (ai,j)
9n,n,
each A
(ai, j) EE~,
(1,...,
n} satisfying
both
{I,
... , n}
both
satisfying
LEMMA
LEMMA
E
jE~
there exist
exist 'IT
there
.{8)
(8)
W1n(A)= min ai,7(i)}
and
and
(9)
(9)
aj,,f(j) =
max {ak,,T(I)}.
E
E
P,n and
and jj EEE
P
PARTITIONING GENERAL
GENERAL PROBABILITY
PROBABILITY MEASURES
MEASURES
PARTITIONING
809
809
PROOF. Since
Since A
A EE~,
n' it
it is
is easy
easy to
to see
see that
that
PROOF.
9w no
Wn(A) == max{
max{ mmin
~ {ai,,T(i)}:
{ai,'IT(i)}: 'ITG
TE
E
Wn(A)
l~l~n
Pn} .•
Pn
Let 7r
'IT * EE Pn
Pn satisfy
satisfy (10)
(10) and
and (11),
(11),
Let
(10)
(10)
~(A) =
= mln
~nfai,*(i)}l
{ai, 'IT*(i)} ,
Wn(A)
1?i~n
l~l~n
n
(11)
(n)
n
= max
E a*a(i)
aide,(s):
itlai,,,.(i)=maxCtlai,,,(i):
(in ai (i)) =
'lTEPn'l~n{ai,"(i)}
,
7tEPEmP
=
Wn(A)
Wn(A)}.
Renumbering if
if necessary,
necessary, assume
assume 'IT*(i)
= ii for
for all
all ii == 1,
... , n,
n, and
and
1,...,
7T*(i)=
Renumbering
will
that
=
now
be
shown
It
<
<
Wn(A)
=
al,1
~
a
,2
~
•••
~
an,n.
It
will
now
be
shown
that
*** an,
Wn(A) a1,1 < a2,2
2
{1,....
..,,n},
somejE {1,
n},
(12)
ajj == max ak,
ak i,
j ' for
forsomejE
(12)
a>;
I~k~n
'
which, with
with (10),
(10), will
will complete
complete the
the proof.
proof.
which,
To
establish
(12),
suppose
by
way
of contradiction
contradiction that
that for
for each
each jj E
E {I,
n},
. . .,, n},
(1, ...
To establish(12),supposebywayof
'i,
el
(1...
,
n}
satisfying
(with
maj1j2
< MaX1
... , in
a jja>;<
maxl~-k
n{ak,j}·
Then there
there exist
exist il.
iI' i2,
i 2 , •••
in E {I, ... , n} satisfying (with
< n{ak,
<-k~
j}. Then
i o : = 1)
1)
io:=
< a·
=
W(A)
= a·
. <
.
Wn(A)
ail,io)
aio,io
n
lO' lO
lh lO'
(13)
(13)
< ai
ail, il1 <
ai1,i
ai2,,il.
2 ,i1 '
< a.
ai
an-1
in
in-1
in-1
G {I,
Since iion...,
... , n},
+ I)-tuple
the ordered
ordered(n
contains
(1,...,
(n +
1)-tuple(i
n}, the
in) contains
(io,
in E
o, ••. , in
o, iil,...,
l , ... , in)
k
E
...
,
n
I}
and
k
E
{O,
...
n
and
a primitive
primitive cycle,
E {O,
thereexists
that is,
exists jj E
cycle,that
is, there
1)
{0,...,, nn -- j)j}
{0,...,
=
and
i1
are
distinct
suchthat
that ij'
1
...
,
such
i
+
...
,
i
+
are
distinct
and
i
=
i
+
+
1,
.
ij+k+
j I
j
j k
j k l
) 'i+k
ij, ijj+
= j + m for
GP
m=
=
Next
form
definedby
the permutation
Nextconsider
considerthe
7rE
by if(i
7T(ij+m+i)
permutationif
Pn
j + m + l ) = iij+n
n defined
0,
IT * otherwise.
By
1,...
...,, k,
and == '7T
otherwise.
0,1,
By (13),
(13),
k, and
= a·lj+m+l' lj+m
a·
. > a·lj+m' lj+m
. m> aa ,
W
(A)'
l ,, I =
Wn(A),
=
lj+m+l'
I-('
T lj+m+l ) =
n
1, 'f(ij+.+)
aij+.+
ai,+m+lGi>,ij+a
= 0,
< minl~
the definition
of if
that Wn(A)
m=
...,, k,
so the
definitionof
for m
for
1, ...
7rimplies
impliesthat
Wn(A) ~
01,
k,so
i nai,
naLw(i)'
f(i),
mini i~
(
and
of
A)
that
of W
that
thedefinition
definition
and hence
henceby
by the
Wn(A)
n
=
min ai,w(i).
(14)
(14)
Wn(A)
~
Wn(A) =
ai,i(i).
l::S;;l::s;;n
> Ei=lai,
that Ei=la
of if
7r also
also imply
implythat
EL jai,
#(i)>
But (13)
(13) and
and the
the definition
definitionof
But
Ejai, 'IT*(i)
,f*(i)
i ,w(i)
the
of (12),
which,
(11).
the proof
and the
with(14),
This completes
completesthe
proofof
(12), and
which,with
contradicts
(11). This
(14), contradicts
lemma.
lemma.0EJ
ofaa
The
states
of
an optimal
is always
The next
nextproposition
thereis
alwaysan
optimalpartitioning
partitioning
proposition
statesthat
thatthere
the
in which
the sum
sum of
of the
stochastic
that is,
matrix in
stochastic matrix
which the
the "cooperative
"cooperative value,"
value," that
is, the
partition-assignment
values,
one.
leastone.
is at
at least
partition-assignment
values,is
PROPOSITION 3.3.
3.3. For
For each
A
each A
PROPOSITION
Hm
both
II
m sa~fying
satisfyingboth
(15)
(15)
Wn(A)=
==
is aa partition
thereis
{ JiJnl EEC
partition {Ji}i-=I
(ai,j)
Snmm there
(ai, j) EE Sn,
mi
1-njEJ1
aij}
T. P.
P. HILL
HILL
T.
810
810
and
and
nn
1.
L EL ai,j~
E
aij 2 1.
i=1 jEJi
i=l
JEJ;,
(16)
(16)
= (ai,j)
Fix A
By the
the definition
definition
of W
thereexists
exists aa
PROOF.
m. By
(ai jj) EE Sn,,
PROOF. Fix
A =
Sn,m.
"of
Wn,
n, there
>
A
of
Hm
If
=
)
E
the
(15).
m
n,
let
(di
S,
n
be
partition
satisfying
partition {e!i}i=l
of
II
satisfying
(15).
If
m>
n,
let
A
=
(ai,j)
E
Sn~n
be the
m
{Jjj~'1
= Lk
matrix
definedby
and
observe
that
both
4*E j~ai,
matrix defined
by ai,
j =
k'
and
observe
that
both
k,
ai J
Jai,
(17)
W
(A)
=
(A) =
min a·1.,1.i.
=mn
= WnnA)
(17)
Wn(A)
n
l:s;;i:s;;n ai,
1<i<n
and
and
nn
a··=
(18)
(18)
~ I., I.
Lai~i=
k..J
(19)
=
~-l(A)
Wn1(A) =
nn
ai,
~ i...J
~ a·
L
i...J
I.,}'.
i=1
i=1 jEJ1
i=l
i=l
JEJ;,
establishthe
theproposition
fornn X
n stochastic
stochastic
By (17)
(17) and
and (18),
(18),it
it is
is enough
enoughto
to establish
By
proposition for
X n
< n,
matrices
A (if
A). The
m<
will
matricesA
add nn - m
m columns
columnsof
ofzeros
zerosto
to A).
The proof
proofwill
(if m
n, simply
simplyadd
= 11 the
proceed by
by induction
on
is
induction
on n;
fornn =
theconclusion
is trivial,
conclusion
so assume
proceed
n; for
it holds
holds
trivial,so
assumeit
= (ai,
....,, nn -- 1 and
and let
let A
for1,
2,...
A =
j) EE=Sn,
for
l, 2,
n.
(ai, j)
Sn,n.
(1,...,
(9).
and jj EE {I,
n} satisfying
satisfying
(8) and
and (9).
By Lemma
Lemma3.2
3.2 there
thereexists
exists'ITrEE PFn
By
Pn and
... , n}
(8)
= nn =
= 'IT(n),
if necessary,
Reordering
by (9)
assume jj =
and observe
observethat
the
7T(n),and
that by
(9) the
Reorderingif
necessary,assume
(n
A by
by deleting
matrix A obtained
obtained from
from A
row
deletingthe
the nth
nth row
(n - 1)
1) XX (n
(n -- 1)
1) matrix
is substochastic
withrow
1 - an,n
=
and column
columnis
substochastic
row sums
sums Lj~fai,j
fora.ll
2 1
all ii =
and
with
an, n for
Ej--lAi, j ~
exists
1,
... , n
that
the induction
inductionhypothesis
hypothesis
that there
thereexists
1,...,
n -- 1.
1. It
It follows
followseasily
easilyfrom
fromthe
satisfying
both
ITEe P
w
both
Pn,
n - 1 satisfying
and
and
mm {ai,ii(i)}
~n
i, (i)}
l:S;;t:s;;n-l
n-l
n-1
L
2 1r(i) ~
Eai,ai,ii(i)
(20)
(20)
i=
i=11
an,n·
an,n
= w(i)
= n,
< n
= 'IT(n)
Defining if
by if(i)
for ii <
n and
and if(n)
and (19)
(19)
79(n) =
r(n) =
n, (8)
(8) and
Defining
orEE P
79(i) =
#f(i)for
Pn
n by
together imply
that
together
implythat
= min{
= min
(21)
(21)
~(A) =
min{~_l(A),
an
n}
~n {ai
an,
n) =
Wn-1(A),
Wn(A)
ai,w(i)},
j(i)i
,
l:s;;t:s;;n'
and (20)
and
of
and the
of ifIrimply
implythat
that
(20) and
thedefinition
definition
nn
L
1.
E 'ai,
a, w(i)
>i ~ 1.
(22)
(22)
i=1
i=l
and (22)
The
conclusion
then
from
{7r(i)}
i) }
The induction
induction
follows
from(21)
(21) and
(22) by
bytaking
takingJ;,
conclusion
thenfollows
Ji== {if(
for
i = 1, ... , n. 0
fori=l,...,n.E
Not all
partitions [partitions
achieving
~(A)] satisfy
Not
all optimal
achievingWn(A)]
satisfy(16).
(16).
optimalpartitions
[partitions
EXAMPLE
EXAMPLE3.4.
3.4.
Let
Let
A =
A=
0.3 0.4]
0.4
0.3 0.3
0.3
0.3
0.3
0.4.
0.41.
0.3
0.3
[
0.3 0.4
0.4 0.3
0.3]
L0.3
PARTITIONING
PARTITIONINGGENERAL
GENERAL PROBABILITY
PROBABILITY MEASURES
MEASURES
811
811
= 1,2,3,
{JJ=
= {i},
The partition
{i}, ii =
1,2,3, satisfies
The
partition {~}
~tisfies
{L
= 0.3 == 1min3
It;(A)
~n {
W3(A)=0.3
1~£~3
E ai,j}
ai}j
jE~
= min{al,l'
minn{al,,aa2,2,
a3,3)
=
2 ,2' aa,a},
but XsiE=?
hai,j = 0.9 < 1.
OF THEOREM
THEOREM 1.2.
1.2. That
PROOF
PROOF OF
That Vn{a)
is attained
attainedfor
forall
all aa follows
followsfrom
from
Vn(a) is
= ...
1.1 by
Theorem1.1
takingILl
u1 = IL2
= ILAn
Theorem
by taking
=
A2 =
.
n•
E fJJ{
Fix
(0,1] and
and ILl'
9)(a).
By an
an argument
argument
directlyanalogous
analogousto
to
Fix aa EE (0,1]
... ' IL,Un
a). By
directly
Al...,
n E
in the
that
proof of
be assumed
of Lemma
Lemma2.2,
it may
loss of
that in
the proof
assumedwithout
withoutloss
of generality
2.2,it
maybe
generality
< 2naIn other
that {ILi}'i=l
atomic each
each with
with at
at most
most m
m~
2naa1l atoms.
atoms. In
other
that
purely atomic
are purely
{fiu}in1 are
if
words,
to
it suffices
suffices
to show
showthat
thatif
words,it
(23)
(23)
< aa
A
and ai,j
A == (ai,j)
ai,j ~
Sn,m and
(ai, j) EE Sn,m
for
all i = 1, ... , n and
j = 1, ... , m,
and ]=l,...,m,
foralli=l,...,n
then
then
(24)
(24)
Wn(A) 2 Vn(a).
E II
Fix
3.3 there
thereexists
existsaa partition
Fix A
A satisfying
(23).
By Proposition
3.3
partition {Jj)~'
{~}'i=l E
satisfying
(23). By
Proposition
rIm
m
satisfying
(15)
and
(16).
To
prove
(24),
fix
n
>
1,
k
~
1,
and
and
To
fix
n
>
k
and
satisfying
(15)
(16). prove(24),
1, > 1,
1
= [(k
aa E I(n,
[(n, k)
«k +
+ l)n
+ l)k1)n -- 1)-\(kn
k) =
[(k +
1)-1,(kn -- 1)-1].
1)-1]
1)k-1((k
< 11 -- k{n
of contradiction,
that 'EjEJlai,j
Suppose,
by way
that
Suppose, by
way of
contradiction,
k(n -- l)a.
1)a. By
By (16),
(16),
YjeJjai j <
E
>
so
for
some
i
>
The
Li=2LjEJiai,j>
k{n
1)
a,
so
for
some
i
E
{2,
...
,
n},
"EjEJiai,j>
ka.
The
ka.
{2,...
k(n
-1)a,
,n,
2
ej
j
ai,
Jaij
ni=2y2jej
argument
now
proceeds as
proof of
having
nowproceeds
as in
in the
theproof
ofTheorem
Theorem1.1,
thekey
argument
1.1,the
keydifference
difference
having
= IL22 =
=
= IL n
been
the use
been the
... =
use of
of Proposition
3.3 (which
is trivial
trivialfor
forthe
Proposition3.3
(whichis
the ILl
A1 =
n
context
contextof
ofTheorem
Theorem1.1).
1.1).00
4.
proposition is
is an
4. Several
an
remarks concerning
The following
Several remarks
concerningV
following
proposition
V.(a).
n ( «). The
easy
of
ofthe
thedefinition
definition
of Vn{a).
easy consequence
consequenceof
Vn(a).
PROPOSITION 4.1.
For each
and
on
on
PROPOSITION
4.1. For
1, V
is continuous
continuous
and nonincreasing
each nn ~
2 1,
nonincreasing
Vn(
n{·)) is
[0,1],
piecewise linear
linearon
on (0,1],
and satisfies
[0,1], piecewise
(0,1], and
satisfies
(i)
(i)
(ii)
(ii)
l
= nVn(O)
,
n-1,
Vn(O) =
= 0;
V
n(l) =
VK(M)
> 0,
Vn+l(a)
if Vn(a)
Vn+(a) << Vn(a),
Vn(a) , if
Vn(a) >
= Vn(a),
Vn+l(a)
Vnj?(a) =
and
and
(iii)
(iii)
and
and
= 0;
if Vn(a) =
0;
if
~Vn(a)> n-1 -(n - )n-la
The
points at
endpoints
The critical
criticalpoints
is
at the
the left-hand
left-hand
endpointsof
of the
the intervals
intervalswhere
whereV
Vn
n is
constantare
are local
local minima.
minima.For
For example,
has local
at 1/3,
constant
local minima
minimaat
example,~
V2 has
1/3, 1/5,
1/5,
and for
forthe
the first
in the
1/7,
... ; and
is
firstof
of these,
one interpretation
is that
case of
of
1/7,...;
these,one
interpretation
that in
the case
T. P.
P. HILL
HILL
T.
812
812
bisection (n
(n =
= 2),
2), atoms
atoms of
of mass
mass exactly
exactly 1/3
1/3 are
are locally
locally the
the worst-in
worst-in general
general
bisection
atoms slightly
slightly less
less than
than or
or slightly
slightly greater
greater than
than 1/3
1/3 allow
allow better
better partitions.
partitions.
atoms
5. Applications
Applications to
to L1
L 1 spaces
spaces and
and statistical
statistical decision
decision theory
theory It
It is
is easy
easy
5.
to translate
translate the
the settings
settings of
of Theorems
Theorems 1.1
1.1 and
and 1.2
1.2 to
to the
the theory
theory of
of L1
L l spaces;
spaces; the
the
to
next theorem
theorem is
is the
the analog
analog of
of Theorem
Theorem 1.2.
1.2.
next
THEOREM
THEOREM 5.1.
5.1. Suppose
Suppose X
A is
is aa Borel
Borel measure
measure on
on R.
IR. If
If fl' f2,...,
f2' ... ' fn
f n eE L1(X)
Ll(A)
1l,
satisfy
satisfy
(i) fi0Oi=1,...,n;
fi ~ 0, i = 1, ... , n;
(i)
If
= 1,
1, ii =
= 1,...,
1, ... , n;
n; and
and
JfidX
i dA =
<
EiR,
A({x})fi(x)
~
a,
for
all
x
E IR,
for
all
x
a,
X({x})fi(x)
(ii)
(ii)
(iii)
(iii)
then there
there exists
exists aa measurable
measurable partition
partition {Ai}i=l
of Ri
IR satisfying
satisfying
then
{Ai}in'1of
f fitidA
= 1, ... , n.
dA ~ Vn(a),
forall i =1,...,n.
Vn(a), toraUi
Ai
Moreover, this
this bound
bound is
is best
best possible,
possible, and
and is
is attained
attained for
for all
all aa and
and n.
n.
Moreover,
statisticaldecision
1.2 to
The final
final theorem
theorem is
is an
an application
application of
of Theorem
Theorem 1.2
to statistical
decision
The
[2]
in [2]
theory
which
is
related
to
similar
applications
of
partitioning
inequalities
inequalitiesin
theorywhichis relatedto similarapplicationsofpartitioning
and
[3].
and [3].
known
theknown
has one
oneof
ofthe
X which
~-valued random
random variable
variable X
Suppose
there is
whichhas
is an
an 9-valued
Supposethere
observation
A
single
one).
knownwhich
whichone). A single observation
is not
not known
ILl'
••• ' ,,Un
IL n (but
distributions
it is
(but it
bu...,
distributions
thedistributions
distributions
ofthe
fromwhich
whichof
X(
w) of
is to
to be
be guessed
thenit
it is
guessedfrom
X is
is made,
and then
of X
made,and
X(X)
a
(measurable)
is
simply
rule
A
decision
ILl'
came. A decision rule is simply a (measurable)
,y •••
observationcame.
the observation
.... ' IL,n
n the
A minimax
minimax
E
distribution
,ut").
then
guess
{Ai}i=l
of
~
("if
X(w)
E
Ai'
then
guess
distribution
IL/').
A
partition
partition{Ai})n'1of Q ("if X(w) Ai,
risk"
R
decision
rule
is
a
partition
which
attains
the
"minimax
risk"
R
given
by
by
the
"minimax
given
attains
which
decisionruleis a partition
= ILi):
= inf{
inf( m~
max p(X~Aildist(X)
0 Aildist(X) =
(25)
... ,lLn)
II~}.
{Ai}
tui): {A
i }7=1 EllI}.
R(uj,...,
(25) R(ILl,
n) =
l~t~n
1<i<n P(X
Since
Since
inf( m~
= inf{
max {1**,,lLn)
R(ILl,
ILi(A i )}: {A
E II~}
{Ai}i }7=1
ie=llI}
{1-tui(Ai)}:
R(y1, ...
pUn)=
l~t~n
1 <i<n
l}
E II~},
{Ai}it1
min {ILi(~~i)}:
1- SUp{
sup( ~n
= 1{ti(A~)} {A
i }7=1 E
l~t~n
Theorem
immediate
consequence.
consequence.
immediate
following
thefollowing
1.2has
has the
Theorem1.2
THEOREM
THEOREM
Then
u •••
9(a).
LetILl'
5.2.
IL nen
E fJJ(
a). Then
5.2. Let
... ,Itt
< 1
1 --Vn(a),
~(a),
R(ILl,
..,' ILn)
An) ~
R(y,uj. ...
n.
all n.
an1i
and all
is attained
attainedfor
all aa and
thisbound
boundis
foraU
and this
zero-sum
ofzero-sum
to the
thetheory
madeto
A
be made
theoryof
also be
can also
[2]) can
(see [2])
A similar
similarapplication
application(see
two-person
games.
games.
two-person
PARTITIONING
PARTITIONING GENERAL
GENERAL PROBABILITY
PROBABILITY MEASURES
MEASURES
813
813
REFERENCES
REFERENCES
How to
[1]
A mer. Math.
Math. Monthly
Monthly 68
[1] DUBINS,
DUBINS, L.
L. and
and SPANIER, E.
E. (1961).
cakefairly.
(1961).How
to cut
cutaa cake
fairly.Amer.
68 1-17.
1-17.
[2]
A. and
[2] DVORETZKY,
DVORETZKY, A.,
A.,WALD, A.
and WOLFOWITZ,
WoLFowITz, J.
J.(1951).
(1951).Relations
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certainranges
ofvector
amongcertain
rangesof
vector
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Pacific J.
Math. 11 59-74.
measures.
J. Math.
59-74.
inequalities
[3]
T. and
and KERTZ, R.
R. (1986).
[3] ELTON, J.,
J., HILL,
HILL, T.
(1986).Optimal-partitioning
fornon-atomic
Optimal-partitioning
inequalitiesfor
non-atomic
probability measures.
Trans.
A mer. Math.
Math. Soc.
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Trans.Amer.
probability
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296703-725.
703-725.
[4] FISHER,
FISHER, R.
R. (1936).
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A mer. Acad.
Acad. Arts
Arts Sci.
(1936). Uncertain
Uncertain
inference.
Proc.
Sci. 71
71 245-257.
245-257.
[5] LYAPOUNOV, A.
A. (1940).
[5]
completement
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BuU. Acad.
A cad. Sci.
(1940).Sur
Sur les
les fonctions-vecteurs
fonctions-vecteurs
complitement
additives.Bull.
Sci.
URSS
4 465-478.
465-478.
URSS 4
[6] NEYMAN,
J. (1946).
[6]
NEYMAN, J.
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C.R.
Acad. Sci.
Paris Sere
A-B 222
(1946). Un
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theoreme
d'existence.
C.R. Acad.
Sci. Paris
Ser. A-B
222843-845.
843-845.
[7]
pragmatique. Econometrica
Econometrica (supplement)
17
[7] STEINHAUS,
STEINHAUS, H.
H. (1949).
la division
(1949).Sur
Sur la
divisionpragmatique.
(supplement)
17 315-319.
315-319.
SCHOOL OF MATHEMATICS
SCHOOL
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INSTITUTE OF TECHNOLOGY
TECHNOLOGY
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