Table Of ContentMEASURES AND INTEGRALS WITHIN CONDITIONAL SET THEORY
ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
Abstract. Ameasureandintegrationtheoryforconditionalsetsisestablished. Asanillustrationof
possibleresults,weproveaconditionalversionoftheCarath´eodoryextensiontheorem,themonotone
convergence theorem, the Fubini theorem, the Radon-Nikody´m theorem, the Daniell-Stone theorem
7
andRiesztyperepresentationtheorems. Classicalfunctionspacesinconditionalsettheoryaredefined.
1
InordertoderivetheresultsanextensionoftheconditionalpowersetasdefinedinDrapeauetal.is
0
proposed. If the complete Boolean algebrais the measurealgebraassociated toa probabilityspace,
2
thenitisshownthatmeasuresontheconditionalBorelσ-algebraofaconditionallycompletedPolish
n space uniquely correspond to probability kernels. The instruments developed in this article may
a provideanalternativetooltostudyprobabilitykernels.
J
0
1
January 11, 2017
]
R
P
. 1. Introduction
h
t
a
m Classicalmeasure andintegrationtheory is basedonthe fact thatthe power setP(X) ofa set X with
[ the operations of intersection ∧ = ∩, union ∨ = ∪ and complement c and the distinguished elements
0=∅ and 1=X is a complete Boolean algebra. A σ-algebra as a plausible domain of a measurement
1
is thus a σ-complete subalgebra of the Boolean algebra P(X). For the foundations of measure theory
v
1 Boolean arithmetic is indispensable. The aim of this article is to establish a measure and integration
6 theorywithinconditionalsettheory[6]. Followingtheclassicaltheorywestartwithaconditionalform
6 of the power set algebra. Due to the inherent locality of conditional sets more care has to be taken to
2
specify conditional subsets.
0
.
1
A conditional set is a collection X of objects x|a where x is an element in a non-empty set X and a is
0
7 an element in a complete Boolean algebra A such that a locality property
1
:
v • if x|b=y|b and a≤b, then x|a=y|a;1
i
X
r and a stability property
a
• if (a ) is a partition of unity in A and (x ) is a family of elements in X, then there exists a
i i
unique element x in X such that x|a =x |a for all i;
i i i
are satisfied. A conditional subset of X is an object of the form C|a which satisfies locally on a the
stability property and is understood to be empty on ac. A conditional measure Φ on a collection X
of conditional subsets is a stable function assigning to C|a locally the value 0 on ac. Therefore the
collectionX mustsatisfysomeformofstabilityproperty. ThestabilityofΦtogetherwiththe stability
of X cannot be realized within the framework introduced in [6]. Therefore we suggest an extension
of the conditional power set in this article. Stable collections in the extended conditional power set
can serve as a domain of conditional measures if they are understood as stable functions. As in [6,
2010 Mathematics Subject Classification. 03C90,60Jxx.
1Recallthata≤bisdefinedbya∧b=a.
1
2 ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
Theorem 2.9], the extended conditional power set can be endowed with the structure of a complete
Boolean algebra.
If the complete Boolean algebra A is the measure algebra associated to a probability space (Ω,F,P),
then some classical function spaces can be related to conditional spaces. For instance, the conditional
realnumbersareconditionallyisometricwiththe spaceofreal-valuedrandomvariablesmoduloalmost
sure equality.2 Given a Polish space X, the space of equivalence classes of random variables with
values in X can be identified with a conditional Polish space on which a conditional Borel σ-algebra
B can be defined by a stable generator of randomly open subsets in X. In Proposition 3.9, it is
proved that a conditional probability measure on B uniquely corresponds to a probability kernel on
the Polish space X. Conditional probability measures exist on the conditional Borel σ-algebra of the
space of equivalence classes of random variables with values in an arbitrary complete metric space. In
this article, we exemplify that for measures in conditional set theory analogues of statements about
measures in classical set theory can be proved. One may use conditional set theory as an alternative
tool to investigate probability kernels.
We discuss the related literature. Conditional set theory is introduced in [6] and it is related to the
conditional analysis in modules [1, 2, 4, 8, 9, 10, 14, 19] whenever the complete Boolean algebra is
an associated measure algebra. Conditional set theory is similar to the Scott-Solovay Boolean-valued
models ofset theory [15, 16]. See for aninvestigationofclassicalvectorlattices from the point view of
Boolean-valued models [12], and the rich bibliography of [12] for further references to Boolean-valued
analysis. An interpretation of differentiation and integration in Boolean-valued models is worked out
in [17]. To the best of our knowledge a systematic treatment of set-theoretic measure and integration
theoryinconditionalsettheorydoes notexist. We referto [5]andits referencesforastudy ofrandom
probability measures on Polish spaces.
Therestofthis paperisorganizedasfollows. InSection2wefix notationsandintroducethe extended
conditional power set and the conditional extended real numbers. Conditional set systems and set
functions and a conditional version of the Carath´eodory extension theorem are presented in Section
3. A conditional form of the Lebesgue integral is derived in Section 4, and there a conditional version
of the monotone convergence theorem and of Fubini’s theorem are proved. In Section 5 we derive
a conditional version of the Radon-Nikody´m theorem. Finally, in Section 6 we prove a conditional
version of the Daniell-Stone theorem which is applied to Riesz representation results on the space of
conditionally continuous functions on the conditionally n-dimensional Euclidean space.
2. Notation and preliminaries
We start with basic concepts from conditional set theory as developed in [6]. We use bold-face letters
to denote conditional sets and plain letters to denote classical sets. Throughout this paper we fix a
complete non-degenerate Boolean algebra A = (A,∧,∨,c,0,1) unless otherwise specified. The supre-
mum and the infimum of a family (a ) = (a ) of elements in A are denoted by ∨a = ∨ a and
i i i∈I i i∈I i
∧a = ∧ a , respectively. We denote by p(1) the collection of all families (a ) of pairwise disjoint
i i∈I i i
elements in A such that ∨a =1.
i
Definition 2.1. A collectionX of objects x|a where x is an element in a non-empty set X and a∈A
is called a conditional set of X (and A) whenever the following properties are satisfied:
(C1) if x|b=y|a, then a=b;
(C2) if x|b=y|b and a≤b, then x|a=y|a;
(C3) if (a )∈p(1) and (x ) is a family of elements in X, then there exists a unique element x in X
i i
such that x|a =x |a for all i.
i i i
2Moregenerally,aconditionalisometrybetweentheconditional completionofametricspaceX andthespaceofall
equivalenceclassesofrandomvariableswithvalues inX isprovedin[7].
MEASURES AND INTEGRALS WITHIN CONDITIONAL SET THEORY 3
Wenametheproperties(C2)and(C3)consistency andstability,respectively. Theobjectx|aisreferred
to as x restricted to a. A concatenation x |a is an element x of X such that x|a = x |a for all i
i i i i i
where (x ) is family of elements in X and (a )∈p(1).
i i
P
Wegiveaconstructionofaconditionalset. LetX beanon-emptysetanddefineX tobethecollection
s
of all families (x ,a ) =(x ,a ) in X ×A such that (a )∈p(1). Identify in X two families (x ,a )
i i i∈I i i i s i i
and (y ,b ) whenever ∨{a :x =z}=∨{b :y =z} for all z ∈X. It can be checked that X defines
j j i i j j s
a conditional set X of objects
s
(x ,a )|a:={(y ,b )∈X : ∨{a : x =z}∧a=∨{b : y =z}∧a for all z ∈X}
i i j j s i i j j
whichiscalledtheconditional setofstepfunctions withvaluesinX. Sincethereexistsabijectionfrom
X to {(x,1): x∈X}⊆X , each family (x ,a )∈X can be written as the concatenation (x ,1)|a .
s i i s i i
Therefore we use the notation x |a for the elements of X . We call the conditional sets N and Q
i i s s s
P
of step functions with values in N and Q the conditional natural and rational numbers, respectively.
P
For a conditional version of basic concepts from measure theory families of conditional subsets of a
conditional set and the conditional set operations of conditional union, conditional intersection and
conditional complement are crucial. We give a detailed construction of these concepts. Let X be a
conditionalsetofX. Wecallanon-emptysubsetC ofX stable if x |a ∈C forallfamilies(x )inC
i i i
and (a )∈p(1). A stable subset C induces a conditional set C of objects x|a where x∈C and a∈A.
i
P
Forinstance,asingleton{x}isastablesubsetofX whichinducestheconditionalsetx={x|a: a∈A}.
A conditional subset of X is a conditional set of the form C|a:={(x|a)|a˜:x|a∈C|a,a˜∈A} where C
is a stable subset of X, C|a:={x|a: x∈C} and
(x|a)|a˜ :={x˜|b∈X: x|a∧a˜=x˜|b∧a˜,ac∧a˜=bc∧a˜}.
Wheneverthereisnoriskofconfusion,wewillwriteC=C|aforaconditionalsubsetofX. Wedenote
by P(X) the collection of all conditional subsets of X.
Let C|a and D|b be conditional subsets of X. We call the conditional set
C|a⊓D|b:=E|c
the conditional intersection of C|a and D|b where c=∨A,
A={c˜: c˜≤a∧b,C|c˜∩D|c˜6=∅}
and
E ={x∈X: x|c∈C|c∩D|c}.
Theconditionalintersectioniswell-definedifcisattainedandE isastablesubsetofX. Weshowthat
cisattained. Bytheaxiomofchoice,chooseawell-orderingonthesetAbymeansofwhichonedefines
a family (c ) of pairwise disjoint elements in A whose supremum is equal to c. For each c there is an
i i
element x in X which restricted to c falls in C|c ∩D|c . The concatenation x |c +x|cc (where
i i i i i i
x may be chosen arbitrarily in X) restricted to c is an element of C|c∩D|c. The previous kind of
P
reasoning is common in conditional set theory, and henceforth referred to as an exhaustion argument.
The conditional set
C|a⊔D|b:=F|a∨b
is called the conditional union of C|a and D|b where
F ={x∈X: x=z |a +z |a +z |(a∨b)c,(a ,a ,(a∨b)c)∈p(1),a ≤a,a ≤b,z ∈C,z ∈D,z ∈X}
1 1 2 2 3 1 2 1 2 1 2 3
is a stable subset of X.
The conditional complement of C|a is defined by
(C|a)⊏ :=G|d∨ac
where
d=∨{d˜: d˜≤a,C|d˜6=X|d˜}
4 ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
and
G={x∈X: x|d˜6=y|d˜∀y ∈C, ∀0<d˜≤d}
is a stable subset of X.
Thesymbol⊑standsfortheconditional inclusion,andcanbedefinedonP(X)byC|a⊑D|bwhenever
C|a⊓D|b=C|a.3 We recall [6, Theorem 2.9].
Theorem 2.2. The structure (P(X),⊓,⊔,⊏,X|0,X) is a complete Boolean algebra.
We introduce the extended conditional power set. Given C|a∈P(X) and a˜∈A, we define
(C|a)|a˜ :={D|b∈P(X): C|a∧a˜=D|b∧a˜, ac∧a˜=bc∧a˜}.
It can be checked that
P(X):={(C|a)|a˜: C|a∈P(X), a˜∈A}
is a conditional set of P(X) which we call the conditional power set of X. Throughout, a stable
collection of conditional subsets ofX is a stable subsetof P(X) with respectto concatenationsformed
in P(X).
In the following paragraphs let X be a conditional set of X and Y be a conditional set of Y. The
conditional product of X and Y is defined as the conditional set
X×Y :={(x,y)|a:=(x|a,y|a): (x,y)∈X ×Y,a∈A}.
Let W|a be a conditional subset of X×Y and x∈X. We define the conditional x-section of W|a by
(W|a) :=C|b
x
where
b=∨ ˜b: ˜b≤a, ∃y ∈Y, ∃(x˜,y˜)∈W such that (x,y)|˜b=(x˜,y˜)|˜b
and n o
C ={y ∈Y : ∃(x˜,y˜)∈W such that (x,y)|b=(x˜,y˜)|b}.
By an exhaustion argument, b is attained, and thus it can be verified that C is a stable subset of Y.
Similarly the conditional y-section of W|a for an element y in Y is defined.
We recall the definition of a stable function and family. A function f: X → Y is called stable if
f( x |a ) = f(x )|a for all concatenations x |a in X. Let C|a be a conditional subset of Y.
i i i i i i
We define the conditional pre-image by f−1(C|a):=D|b where
P P P
b=∨ ˜b: ˜b≤a, ∃x∈X, ∃y ∈C such that f(x)|˜b=y|˜b
and n o
D ={x∈X: ∃y ∈C such that f(x)|b=y|b}.
We define a stable order4 on N by n |a 6 m |b if n ≤ m for all a ∧b > 0. Note that
s i i j j i j i j
{16 m 6 n} is a stable subset of N for every n ∈ N . A stably finite family (x ) in X is the
s s m 16m6n
P P
graph of a stable function {1 6 m 6 n} → X. A stable sequence (x ) in X is the graph of a stable
n
function N →X.
s
The following properties follow from the previous definitions. Let (C ) = (C |a ) be a non-empty
i i i
family of conditional subsets of X, D⊑X and (b )∈p(1). Then one has
i
(S1) ⊓(C |a )=(⊓C )|∧a and ⊔(C |a )=(⊔C )|∨a ,
i i i i i i i i
(S2) ( (C |a )|b )⊏ = (C |a )⊏|b +X|∨(ac∧b ),
i i i i i i i i
(S3) ( C |b )⊓D= (C ⊓D)|b and ( C |b )⊔D= (C ⊔D)|b .
i i i i i i i i
P P
3ForagPeneraldefinitionwPereferto[6,DefinitionP2.8]. P
4See[6,Definition2.15].
MEASURES AND INTEGRALS WITHIN CONDITIONAL SET THEORY 5
For a non-empty family (C ) of conditional subsets of X, a stably finite family (C ) of condi-
ij m 16m6n
tional subsets of X and (b )∈p(1), one has
i
(S4) (⊔ C )|b =⊔ ( C |b ) and (⊓ C )|b =⊓ ( C |b ),
j ij i j ij i j ij i j ij i
(S5) ⊓ C = (⊓ C )|a and⊔ C = (⊔ C )|a wheren= n |a .
P16m6n m 1P≤m≤ni m iP 16m6n m P 1≤m≤ni m i i i
P P P
Let C×D=C|a×D|b and C˜ ×D˜ be conditional subsets of X×Y, (Ci) be a family of conditional
subsets of X×Y, E⊑X×Y and (b )∈p(1). Then we have
i
(S6) C×D⊓C˜ ×D˜ =C⊓C˜ ×D⊓D˜,
(S7) Ci|b = Ci|b for all x∈X,
i x x i
(S8) (E|a) =E |a for all x∈X and a∈A,
(cid:0)P x (cid:1) x P
(S9) C = C |a for all concatenations x |a in X,
Pxi|ai xi i i i
(S10) (E )⊏ =(E⊏) for all x∈X,
x x
P P
(S11) (⊔Ci) =⊔Ci for all x∈X,
x x
(S12) (C|a×D|b) =D|c where c=∨{c˜: c˜≤a, ∃x˜∈C such that x|c˜=x˜|c˜} for all x∈X.
x
We close this preliminary section with basic properties of the conditional extended real line. The
conditional real numbers R areconstructedin [6, Section4]. The conditional extended real numbers R
can be obtained by a conditional completion5 of the conditional set R of step functions with values
s
in R := R∪{±∞}. Denoting by R and R the stable sets inducing R and R, respectively, each x in
R has the form −∞|a +y|a +∞|a where y ∈ R and (a ,a ,a ) ∈ p(1). The conditional addition
1 2 3 1 2 3
and multiplication on R extend to R by stipulating the following conditional version of the classical
conventions. For all −∞|a +x|a +∞|a ,−∞|b +y|b +∞|b ∈R and r =r−|c +0|c +r+|c ∈R
1 2 3 1 2 3 1 2 3
where c =∨{a: r|a<0|a}, c =∨{a: r|a=0|a} and c =∨{a: r|a >0|a}, and r− =r|c +0|cc and
1 2 3 1 1
r+ =r|c +0|cc, set
3 3
(−∞|a +x|a +∞|a )+(−∞|b +y|b +∞|b ):=−∞|(a ∧b )∨(a ∧b )∨(b ∧a )
1 2 3 1 2 3 1 1 1 2 1 2
+0|(a ∧b )∨(a ∧b )+(x+y)|(a ∧b )
1 3 3 1 2 2
+∞|(a ∧b )∨(a ∧b )∨(b ∧a ),
3 3 2 3 2 3
r·(−∞|a +x|a +∞|a ):=−∞|(c ∧a )∨(c ∧a )
1 2 3 3 1 1 3
+r−x|c ∧a +0|c +r+x|c ∧a
1 2 2 3 2
+∞|(c ∧a )∨(c ∧a ).
3 3 1 1
The total order6 on R extends to R via the relation
−∞|a +x|a +∞|a 6−∞|b +y|b +∞|b if b ≤a , a ≤b and x|a ∧b 6y|a ∧b .
1 2 3 1 2 3 1 1 3 3 2 2 2 2
Let R := {x ∈ R: x > 0} and R := {x ∈ R: x > 0} where the relation x > 0 is defined as x > 0
+ ++
andx|a=0|awhenevera=0. NotethatR andR arestablesubsetsofRandR,respectively. For
+ ++
n = n |a ∈ N and a stably finite family (x ) in R, define the stably finite sum and stably
i i s m 16m6n
finite product by
P
x := (x +...+x )|a , x := (x ·...·x )|a ,
m 1 ni i m 1 ni i
16m6n 16m6n
X X Y X
respectively. The collection of all stable open balls7 B (x) := {y ∈ R: d (x,y) < r} where x ∈ R and
r c
r ∈R isastablecollectionofstablesubsetswhichformsaclassicaltopologicalbaseonthesetR,and
+
thus induces a conditional topology8 T on R by [6, Proposition 3.5]. Restricted to R, the conditional
topology T is conditionally equivalent to the conditional Euclidean topology which is defined in [6,
5See[7,Theorem2.1].
6See[6,Definition2.15].
7dc isastablemetricwithvaluesinRinducedbythearctan-metriconR,see[7,Section2].
8See[6,Definition3.1].
6 ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
Section 4]. For a stable sequence (x ) in R , we define the stably infinite series x as the limit
n + n>1 n
of the stable sequence of stably finite sums x .
16m6n m P
P
3. Conditional measure spaces and a conditional version of Carath´eodory’s
extension theorem
In this section we introduce a conditional version of measurability. We prove a conditional version
of Carath´eodory’s extension theorem by means of which we construct a conditional n-dimensional
Lebesgue measure.
Definition 3.1. LetXbeaconditionalset. AstablecollectionX ofconditionalsubsetsofXis called
• a conditional ring whenever C⊓D⊏,C⊔D∈X for all C,D∈X;
• a conditional Dynkin system whenever X ∈ X, C⊏ ∈ X for all C ∈ X and ⊔C ∈ X for all
n
stable sequences (C ) of conditionally pairwise disjoint9 conditional sets in X;
n
• a conditional σ-algebra whenever X ∈ X, C⊏ ∈ X for all C ∈ X and ⊔C ∈ X for all stable
n
sequences (C ) of conditional sets in X.
n
A pair (X,X) is called a conditional measurable space if X is a conditional set and X is a conditional
σ-algebra. Let (X,X) and (Y,Y) be conditional measurable spaces. A stable function f: X → Y is
called conditionally measurable if f−1(C)∈X for all C∈Y.
For a stable collection E of conditional subsets of X, it can be checked that
Σ(E):=∩{X: E ⊆X, X conditional σ-algebra},
∆(E):=∩{X: E ⊆X, X conditional Dynkin system}
are a conditional σ-algebra and Dynkin system called the conditional σ-algebra and Dynkin system
generatedby E, respectively. If E is a conditional topology on X, then the conditional σ-algebra Σ(E)
is calledthe conditional Borel σ-algebra of X.10 We have the following conditionalversionof Dynkin’s
π-λ theorem.
Theorem 3.2. Let Xbeaconditional set andE beastable collection ofconditional subsetsof Xwhich
is closed under stably finite conditional intersections. Then one has
Σ(E)=∆(E).
Proof. The proof is similar to the classical proof by applying Theorem 2.2. (cid:3)
Definition 3.3. Let X be a conditional set and X a conditional ring. A stable function Φ:X →R
+
is called a conditional pre-measure whenever
(M1) Φ(C|a)=Φ(C)|a+0|ac for all C in X, a∈A;
(M2) Φ(⊔C )= Φ(C ) for all stable sequences (C ) of conditionally pairwise disjoint condi-
n n>1 n n
tional sets in X with ⊔C ∈X.
n
P
A stable function Φ:X →R satisfying property (M1) and
+
9Astable sequence (Cn)isconditionally pairwise disjoint ifCn⊓Cm =X|0whenever n isconditionally not equal
tom,i.e.n|a=m|aimpliesa=0.
10A conditional topology is defined in [6, Definition 3.1] to be a conditional collection of conditional subsets in the
conditional setP(X) (see[6,Definition2.7])whichintheextended conditional powersetP(X)canbeseenasastable
collectionofconditionalsubsets.
MEASURES AND INTEGRALS WITHIN CONDITIONAL SET THEORY 7
(M2)′ Φ(⊔ C ) = Φ(C ) for all stably finite families (C ) of conditionally
16m6n m 16m6n m m 16m6n
pairwise conditional sets in X,
P
is called a conditional content. If the domain of a conditional pre-measure is a conditional σ-algebra,
then it is called a conditional measure and the triple (X,X,Φ) a conditional measure space. A condi-
tional measure Φ is conditionally finite if Φ(X)<∞, and conditionally σ-finite if there exists a stable
sequence (D ) in X with D ⊑D whenever m6n and ⊔D =X such that Φ(D )<∞ for all n.
n m n n n
Let Φ be a conditional content on (X,X). By inspection, one has
(M3) Φ(C⊔D)+Φ(C⊓D)=Φ(C)+Φ(D) for all C,D in X;
(M4) Φ(C)6Φ(D) for all C,D in X with C⊑D;
(M5) Φ(D⊓C⊏)=Φ(D)−Φ(C) for all C,D in X with C⊑D and Φ(C)<∞;
(M6) Φ(⊔ C )6 Φ(C )forallstablesequences(C )ofconditionalsetsinX with⊔ C ∈
n>1 n n>1 n n n>1 n
X;
P
(M7) Φ isa conditionalpre-measureifandonly iffor allstable sequences(C )ofconditionalsets in
n
X such that C ⊑C whenever m6n and ⊔C =C for some C in X, one has limΦ(C )=
m n n n
Φ(C);
(M8) If Φ(C) < ∞ for all C in X, then Φ is a conditional pre-measure if and only if for all stable
sequences (C ) of conditional sets in X such that C ⊑ C whenever m 6 n and ⊓C = C
n n m n
for some C in X, one has limΦ(C )=Φ(C).
n
Definition 3.4. Given a conditional set X, a stable function Φ∗: P(X)→ R is called a conditional
+
outer measure whenever
(A1) Φ∗(C|a)=Φ∗(C)|a+0|ac for all C in P(X), a∈A;
(A2) Φ∗(C)6Φ∗(D) for all C,D in P(X) with C⊑D;
(A3) Φ∗(⊔ C )6 Φ∗(C ) for all stable sequences (C ) in P(X).
n>1 n n>1 n n
P
A conditional subset C of X is said to be conditionally Φ∗-measurable whenever
Φ∗(D⊓C)+Φ∗(D⊓C⊏)=Φ∗(D)
for all conditional subsets D of X. Let X(Φ∗) denote the stable collection of all conditionally Φ∗-
measurable conditional sets.
Similarlytotheclassicalcase,onecanprovethatX(Φ∗)isaconditionalσ-algebraandΦ∗ restrictedto
X(Φ∗) is a conditional measure. This provides for the following conditional version of Carath´eodory’s
extension theorem.
Theorem 3.5. Let X be a conditional set, X a conditional ring on X and Φ:X →R a conditional
+
pre-measure. Then there exists a conditional measure Ψ: Σ(X)→R which coincides with Φ on X.
+
Proof. For a conditional subset C of X, put
a :=∨{a: ∃ a stable sequence (D ) in X such that C|a⊑⊔D }.
C n n
By an exhaustion argument, a is attained. Thus Φ∗: P(X)→R given by
C +
Φ∗(C):=inf Φ(D ): (D ) stable sequence in X such that C|a ⊑⊔D a +∞|ac .
n n C n(cid:12) C C
nX>1 (cid:12)
(cid:12)
isawell-definedstablefunction. Itcanbechecked,similarlytotheclassicalcase,thatΦ(cid:12)∗isaconditional
(cid:12)
outer measure. (cid:3)
8 ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
As for the uniqueness of the extensionin the previous theorem, one has the following result which can
be proved by using Theorem 3.2.
Proposition 3.6. Let (X,X,Φ) and (X,X,Ψ) be conditional measure spaces, E a stable collection of
conditional subsets of X which is closed under stably finite conditional intersections and such that X =
Σ(E). Further there exists a stable sequence (D ) in E with ⊔D =X such that Φ(D )=Ψ(D )<∞
n n n n
for all n. If one has Φ(C)=Ψ(C) for all C∈E, then we have Φ(C)=Ψ(C) for all C∈X.
We give two examples of conditional measures.
Examples 3.7. (i) LetX be a conditionalsetofX, X a conditional σ-algebraonX andx∈X. The
conditional Dirac measure centered at x is defined as
δ (C):=1|a+0|ac, C in X,
x
where a=∨{a˜: ∃x˜∈C such that x|a˜ =x˜|a˜}.
(ii) For n = n |a ∈ N , we denote by Rn the conditionally n-dimensional Euclidean space. An
i i s
element x in the stable set Rn has the form x = (x ) := (x ,...,x )|a with (x ,...,x )
P m 16m6n i1 ini i 1 ni
being an element of the product Rni, each i. For x,y in Rn let x6y be defined by x 6y for all
im im
P
1 ≤ m ≤ n and i. Given x 6 y, we define the stable subset [x,y] := {z ∈ Rd: x 6 z 6 y}. It can be
i
checked that
X ={⊔ [x ,y ]|a: ([x ,y]) stably finite, conditionally pairwise disjoint, a∈A}
16l6k l l l l 16l6k
is a conditional ring. By a conditional version of the Heine-Borel theorem [6, Theorem 4.6] and (M8),
the function Λ: X →R given by
+
Λ(⊔ [x ,y ]|a):= (y −x )|a+0|ac
16l6k l l lm lm
16l6k16m6n
X Y
isaconditionalpre-measure. ByTheorem3.5andProposition3.6,Λuniquelyextendstoaconditional
measure on the conditional Borel σ-algebra Bn :=Σ(X) which is called the conditional n-dimensional
Lebesgue measure.
In the following we will give a canonical construction of conditional measures induced by probability
kernels on Polishspaces where the complete Booleanalgebrais associatedto a σ-finite measure space.
3.8. Probability kernels and conditional measures. For the remaining part of this section A
denotes the measure algebra associated to a fixed complete σ-finite measure space (Ω,F,µ). Recall
thatAisobtainedfromF byquotientingouttheσ-idealofµ-nullsets. By[13,Chapter22,Proposition
2.8], the Boolean algebra A is σ-complete and satisfies the countable chain condition, and thus is
complete [18]. We denote the equivalence classes in A by a = [A] where A ∈ F. We write 1 for
A
the characteristic function of A ∈ F. Let L0 be the collection of all measurable functions x: Ω → R
where two of them are identified if they agree µ-almost everywhere, and denote by L0 the subset of
L0 consisting of finite-valued functions. Equalities and inequalities between measurable functions are
understood in the µ-almost everywhere sense. The set L0 induces a conditional set L0 of objects
x|a:={y ∈L0: x1 =y1 for some A∈a} (3.1)
A A
where x∈L0 and a∈A. We denote by L0 the conditional subset of L0 induced by the stable subset
L0. By the countable chain condition, the elements in p(1) have at most countably many non-zero
entries. For (a ) ∈ p(1) and (x ) a sequence of elements in L0 the concatenation x |a is the
n n n n
measurable function x 1 where a =[A ] each n. By adapting the proof of [6, Theorem4.4],
n≥1 n An n n P
one shows that L0 is conditionally isometrically isomorphic to R. The stable order on R corresponds
P
to the order of almost everywhere dominance on L0. We identify R , N and Q with the stable sets
+ s s
of all R -, N- and Q-valued measurable functions, respectively.
+
MEASURES AND INTEGRALS WITHIN CONDITIONAL SET THEORY 9
Recall that given a Polish space S with its Borel σ-algebra S, a probability kernel from (Ω,F) to
(S,S) is a function κ: Ω × S → R such that κ(ω,B) is measurable in ω for fixed B ∈ S and
+
a probability measure in B for fixed ω ∈ Ω. The following result relates probability kernels with
conditionalprobabilitymeasuresontheconditionalBorelσ-algebraoftheconditionalcompletion11S .
c
Denoting by L0(S) the space ofallBorelmeasurablefunctions x: Ω→S modulo µ-almosteverywhere
equality,by a copy of the proofof [7, Theorem4.1], S is conditionally isometrically isomorphicto the
c
conditional set L0(S) where L0(S) is defined as in (3.1). By [11, Theorem A1.2], there exists a Borel
isomorphism from S to a Borel subset of R. Therefore it is enough to prove the following assertion
where B denotes the Borel σ-algebra of R and B =B1 the conditional Borel σ-algebra of L0.
Proposition 3.9. To every probability kernel κ: Ω×B → R is uniquely associated a conditional
+
probability measure Φ : B →L0 and to every conditional probability measure Φ: B →L0 is uniquely
κ + +
associated a probability kernel κ : Ω×B → R such that the identities Φ = Φ and κ = κ hold
Φ + κΦ Φκ
almost everywhere.
Proof. Let κ: Ω×B→R be a probability kernel. For x,y ∈Q with x6y there exists (a )∈p(1)
+ s n
andasequence(x ,y )inQ2 withx ≤y eachnsuchthat[x,y]= [x ,y ]1 wherea =[A ]
n n n n n≥1 n n An n n
each n. On
P
X ={⊔ [x ,y]|a: ([x ,y ]) stably finite, conditionally pairwise disjoint, x ,y ∈Q ,a∈A}
16l6k l l l l 16l6k l l s
define Φ by
κ
Φ (⊔ [x ,y ]|a):= κ(ω,[x ,y ])1 (ω) 1
κ 16l6k l l lm lm Alm An∩A
!!
nX≥1 1≤Xl≤kn mX≥1
where k = k 1 and [x ,y ] = [x ,y ]1 . By Theorem 3.5 and Proposition 3.6,
n≥1 n An l l m≥1 lm lm Alm
Φ uniquely extends to a conditional probability measure on B. Conversely, let Φ: B → L0 be a
κ P P +
conditional probability measure. For each q ∈Q define f(·,q): Ω→[0,1] by f(ω,q)=Φ((−∞,q])(ω).
By(M4),onehasf(·,p)6f(·,q)wheneverp≤q. ThereforeA={ω ∈Ω: f(ω,q) increasing in q ∈Q}
is measurable. For each ω ∈ A the function f(ω,q) is increasing in q with limits 0 and 1 at −∞ and
+∞, respectively. Set
F(ω,x):=1A(ω)inf f(ω,q)+1Ac(ω)1{x≥0}, x∈R, ω ∈Ω.
q>x
By(M4)and(M8),F(ω,·)isadistributionfunctionforeveryω ∈Ω,andthusthereexistsaprobability
measureκ(ω,·)suchthatκ(ω,(−∞,x])=F(ω,x)forallx∈Randω ∈Ω. Foreveryx∈Rthefunction
F(ω,x) is measurable in ω. By a monotone class argument κ: Ω×B → R is a probability kernel.
+
Using a monotone class argument based on an almost everywhere interpretation of (M5) and (M7)
shows that κ(·,B) = Φ(B) for all B ∈ B where B denotes here the conditional Borel subset of L0
induced by the stable set 1 B. Everyother kernelκ′ coincideswith κ on the collectionofsets (−∞,q]
Ω
for all q ∈ Q which yields uniqueness due to a monotone class argument. The reciprocality identities
are immediate from the constructions. (cid:3)
4. A conditional Lebesgue integral and a conditional version of Fubini’s theorem
For the sake of simplicity, let (X,X,Φ) be a finite conditional measure space throughout this section.
We introduce a stable indicator. For C=C|a∈X and x∈X, define
a :=∨{a˜: a˜≤a, ∃x˜∈C such that x|a˜=x˜|a˜}.
x
Since a = ∨(a ∧a ) for all concatenations x |a in X, the function 1 : X → R defined by
Pxi|ai xi i i i C
1 (x) :=1|a +0|ac is stable, and it is called the stable indicator of C. The following properties can
C x x P
be checked from the definition:
11See[6,Section2].
10 ASGARJAMNESHAN,MICHAELKUPPER,ANDMARTINSTRECKFUSS
(D1) 1 =1 |a+0|ac for all C∈X and a∈A,
C|a C
(D2) 1 = 1 |a for all concatenations C |a in X,
PCi|ai Ci i i i
(D3) 1 = 1 for all stable sequences (C ) of conditionally pairwise disjoint elements
⊔n>1Cn Pn>1 Cn P n
of X,
P
(D4) 1 =1−1 for all C in X.
C C⊏
We define a conditionally simple function. A conditionally measurable function f: X → R is called
+
conditionally simple if there exist a stably finite family (x ) in R and a stably finite family
m 16m6n +
(C ) of conditionally pairwise disjoint elements in X such that f = x 1 . By a
m 16m6n 16m6n m Cm
straightforwardadaptationof the classicalconstruction,one can show that eachconditionally measur-
P
able function f: X →R is the pointwise limit of a stable increasing sequence of conditionally simple
+
functions. We define the conditional Lebesgue integral in three steps. First, for a conditionally simple
function f = x 1 , define
16m6n m Cm
P
fdΦ:= x Φ(C ).
m m
ZX 16m6n
X
Second, for a conditionally measurable function f: X →R , define
+
fdΦ:=sup f dΦ
n
ZX ZX
where (f ) is a stable increasing sequence converging to f. Third, for a conditionally measurable
n
function f: X →R define
fdΦ:= f+dΦ− f−dΦ
ZX ZX ZX
where f+ = max{f,0} and f− = max{−f,0}, and we call f conditionally integrable whenever
|f|dΦ<∞ where |f|=f++f−.
X
R
We introduce conditional Lp-spaces. Let f: X → R be conditionally measurable. For 1 6 p < ∞,
define the conditional p-th norm of f by
1/p
kfk := |f|pdΦ .
p
(cid:18)ZX (cid:19)
For p=∞, define the conditional essential supremum norm of f by
kfk :=inf{r>0: Φ({|f|>r})=0}.
∞
For arbitrary 1 6 p 6 ∞ with p = p˜|a+∞|ac where p˜ < ∞, we denote by Lp the collection of all
conditionally measurable functions f: X →R such that
kfk :=kfk |a+kfk |ac <∞,
p p˜ ∞
where two of them f and f˜are identified whenever Φ({f 6=f˜})=0 where {f 6=f˜}:=C|a,
a=∨ a˜: ∃x∈X such that f(x)|˜b6=f˜(x)|˜b for all 0<˜b≤a˜
n o
and
C = x∈X: f(x)|a˜ 6=f˜(x)|a˜ for all 0<a˜≤a .
It can be checked that each k·nk (1 6 p 6 ∞) is a stable function whichoinduces a conditional norm
p
on Lp in the sense of [6, Definition 5.11]. By following the classical arguments,each Lp endowed with
the conditional p-th norm is a conditional Banach space.12
As in the classical case, we have a conditional version of the monotone convergence theorem.
12Aconditional Banachspaceisdefinedintheparagraphfollowing[6,Definition5.11].
