A CATEGORY OF PROBABILITY SPACES AND
MONETARY VALUE MEASURES
TAKANORI ADACHI AND YOSHIHIRO RYU
Abstract. We introduce a category Prob of probability spaces
whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving direction
of information with a way of its interpretation. We introduce a contravariant functor E from Prob to Set, the category of sets. The
functor E provides conditional expectations along arrows in Prob,
which are generalizations of the classical conditional expectations.
For a Prob-arrow f − , we introduce two concepts f − -measurability
and f − -independence and investigate their interaction with conditional expectations along f − . As an application of the category
Prob, we define monetary value measures as contravariant functors from Prob to Set.
1. Introduction
One of the most prominent trials of applying category theory to probability theory is Lawvere and Giry’s approach of formulating transition
probabilities in a monad framework ( [Lawvere, 1962], [Giry, 1982] ).
However, there are few trials of making categories consisting of all
probability spaces due to a difficulty of finding an appropriate condition of their arrows. One of the exceptions is a way to adopt measurepreserving functions as arrows. With this setting, for example, Franz
develops a stochastic independence theory in [Franz, 2003]. Our approach is one of this simple-minded trials. We introduce a category
Prob of all probability spaces in order to see a possible generalization of some classical tools in probability theory including conditional
expectations. Actually, [Adachi, 2014] provides a simple category for
formulating conditional expectations, but its objects and arrows are so
limited that we cannot use it as a foundation of categorical probability
theory.
The heart of this paper is a definition of arrows of Prob that are
maps corresponding to measurable functions satisfying an absolutely
Date: May 15, 2017.
2000 Mathematics Subject Classification. Primary 60A99, 16B50; secondary
60G20, 18B99 .
Key words and phrases. conditional expectation, Radon-Nikodym derivative,
category theory .
This work was supported by JSPS KAKENHI Grant Number 17K01248.
1
2
T. ADACHI AND Y. RYU
continuous requirement that is weaker than the measure-preserving requirement (Section 2). The resulting Prob-arrow can be seen as an
evolving direction of information with a way of its interpretation. We
will see that the requirement allows us to extend some important notions relativized by a σ-algebra in classical probability theory to notions
relativized by a Prob-arrow f − . For example, we introduce notions of
a conditional expectation along f − (Section 3), a f − -measurable function (Section 4) and a random variable independent of f − (Section 5).
Those are considered as generalizations of the classical counterparts.
The existence of those natural generalizations may support a claim
saying that the requirement for Prob-arrows is a natural one. As
an application of the category Prob, we generalize the notion of monetary value measures developed in [Adachi, 2014] by extending their
base category to Prob (Section 6).
Actually, the category Prob and functors developed in this paper
convey more natural and richer structures than those introduced in
[Adachi, 2014].
2. Category of Probability Spaces
In this paper, X̄ = (X, ΣX , PX ), Ȳ = (Y, ΣY , PY ) and Z̄ = (Z, ΣZ , PZ )
are probability spaces.
Definition 2.1. [Null-preserving functions] A measurable function f :
Ȳ → X̄ is called null-preserving if f −1 (A) ∈ NY for every A ∈ NX ,
−1
where NX := P−1
X (0) ⊂ ΣX and NY := PY (0) ⊂ ΣY .
The following characterization is straightforward.
Proposition 2.2. Let f : Ȳ → X̄ be a measurable function. Then, f
is null-preserving iff PY ◦ f −1 ≪ PX .
The following diagram (that does not commute in general) might be
helpful to see the situation we consider.
Y
NO Y
⊂
f −1
f −1
f
X
NX
ΣO Y L
LLP
LLY%
⊂
ΣX
rr9
rrr
[0, 1]
PX
Proposition 2.3. If two functions g : Z̄ → Ȳ and f : Ȳ → X̄ are
null-preserving, then so is f ◦ g : Z̄ → X̄.
Proof. Immediate.
□
Proposition 2.3 makes the following definition well-defined.
Definition 2.4. [Category Prob] A category Prob is the category
whose objects are all probability spaces and the set of arrows between
A CATEGORY OF PROBABILITY SPACES
3
them are defined by
Prob(X̄,Ȳ ) := {f − | f : Ȳ → X̄ is a null-preserving function.},
where f − is a symbol corresponding uniquely to a function f .
We write IdX for an identity measurable function from X̄ to X̄,
while writing idX for an identity function from X to X. Therefore, the
identity arrow of a Prob-object X̄ is Id−
X.
An arrow f − in Prob can be considered to represent an evolving
direction of information with a way of its interpretation. The information is evolving along f −1 but with a restriction to its accompanying
probability measure.
In order to see it more concretely, let us consider a case when f
is an identity function on X, that is, consider a Prob-arrow id−
X :
(X, Σ1 , P1 ) → (X, Σ2 , P2 ). Then, we have Σ1 ⊂ Σ2 and P2 = P2 ◦
id−1
X ≪ P1 . This means that the information is growing while the
support of the probability measure is decreasing. The latter makes
sense if we think a situation: someone believed that some event among
many other events may happen, but now she has changed her mind to
believe that the event will never occur, and so she can concentrate on
other possible events.
Actually, [Adachi, 2014] treats this special situation. It introduces,
for a given measurable space (Ω, G), a category χ(Ω, G) whose objects
are all the pairs of the form (FU , PU ) where FU is a sub-σ-field of G
and PU is a probability measure on G. And it has a unique arrow from
(FV , PV ) to (FU , PU ) only when FV ⊂ FU and PU ≪ PV . Note that
there is a natural embedding ι of the category χ(Ω, G) into Prob.
χ(Ω, G)
ι
/ Prob
(FV , PV )
ι
/ (Ω, FV , PV | FV )
∗
(FU , PU )
ι(∗):=id−
Ω
ι
/ (Ω, FU , PU | FU )
Proposition 2.5. A probability space 0 := ({∗}, {{∗}, ∅}, P0 ), where
P0 ({∗}) := 1 and P0 (∅) := 0, is an initial object of the category Prob.
Actually, for a probability space X̄, !−
X : 0 → X̄ is a unique arrow in
Prob, where !X : X → {∗} is a function such as !X (x) = ∗ for all
x ∈ X.
In the following discussions, we fix the state space be a measurable
space (R, B(R)) for a simplicity. L∞ (X̄) is a vector space consisting of
R-valued random variables v such that PX - ess supx∈X |v(x)| < ∞, while
L1 (X̄)
∫ is a vector space consisting of R-valued random variables v such
that X |v| dPX has a finite value. For two random variables u1 and u2 ,
we write u1 ∼PX u2 or u1 = u2 PX -a.s. when PX (u1 ̸= u2 ) = 0.
4
T. ADACHI AND Y. RYU
L∞ (X̄) and L1 (X̄) are quotient spaces L∞ (X̄)/ ∼PX and L1 (X̄)/ ∼PX ,
respectively.
Proposition 2.6. Let u1 and u2 be two elements of L∞ (X̄), and f − be
an arrow in Prob(X̄, Ȳ ). Then, u1 ∼PX u2 implies u1 ◦ f ∼PY u2 ◦ f .
Proposition 2.6 makes the following definition well-defined.
Definition 2.7. [Functor L] A functor L : Prob → Set is defined by:
XO
X̄ L
f−
f
:=
L∞ (X̄)
∋
[u]∼PX
_
Lf −
Y
/ LX̄
Ȳ L
/ LȲ
Lf −
:=
L∞ (Ȳ )
∋
[u ◦ f ]∼PY
3. Conditional Expectation Functor
Theorem 3.1. Let f − be an arrow in Prob(X̄, Ȳ ). Then, for any
v ∈ L1 (Ȳ ) there exists a u ∈ L1 (X̄) such that for every A ∈ ΣX
∫
∫
(3.1)
u dPX =
v dPY .
A
f −1 (A)
Moreover, u is determined uniquely up to PX -null sets. In other words,
if there are two u1 , u2 ∈ L1 (X̄) both satisfying (3.1), then PX (u1 ̸=
u2 ) = 0 or u1 ∼PX u2 .
−
We write a version of this u by E f (v), and call it a conditional
expectation of v along f − . Therefore,
∫
∫
f−
(3.2)
E (v) dPX =
v dPY .
A
f −1 (A)
Proof. Let us prove the uniqueness first. Assume that both
u1 and
∫
1
u
∈
L
(
X̄)
satisfy
(3.1),
and
that
P
(u
=
̸
u
)
>
0.
Then
u
dPX =
X
1
2
A 1
∫2
u dPX . Now let w := u1 − u2 . Then PX (w > 0) + PX (w < 0) =
A 2
PX (w ̸= 0) > 0. Therefore PX (w > 0) > 0 or PX (w < 0) > 0. In
the former case, there exists n ∈ N such that PX (w > n1 ) > 0 since
{w > 0} =↑ lim{w > n1 } ∫and the monotone convergence property of
measures. But, then 0 = {w> 1 } w dPX ≥ n1 PX (w > n1 ) > 0, which is
n
a contradiction. Hence PX (w > 0) = 0. A similar discussion works for
the latter case. Therefore, PX (u1 ̸= u2 ) = 0.
1
∗
∗
∫ Now, for v ∈ L (Ȳ ), define∗ a function v : ΣY → R by v∗ (B) :=
v dPY for B ∈ ΣY . Then, v is a signed measure satisfying v ≪ PY .
B
Therefore, v ∗ ◦ f −1 ≪ PY ◦ f −1 ≪ PX
theorem,
∫ . So by Radon-Nikodym
∫
there exists u ∈ L1 (X̄) such that A u dPX = A d(v ∗ ◦ f −1 ) for any
A ∈ ΣX , where u is unique up to PX -null sets. Hence
∫
∫
∫
∗
−1
∗
−1
∗
−1
u dPX =
d(v ◦f ) = (v ◦f )(A) = v (f (A)) =
v dPY .
A
A
f −1 (A)
A CATEGORY OF PROBABILITY SPACES
5
□
Id−
X
Proposition 3.2. For u ∈ L1 (X̄), E (u) ∼PX u.
∫
∫
∫
−
Proof. For every A ∈ ΣX , A E IdX (u) dPX = Id−1 (A) u dPX = A u dPX .
X
□
Proposition 3.3. Let f − and g − be arrows in Prob like:
X̄
f−
/ Ȳ
g−
/ Z̄ .
−
−
(1) For v1 , v2 ∈ L1 (Ȳ ), v1 ∼PY v2 implies E f (v1 ) ∼PX E f (v2 ).
−
−
−
−
(2) For w ∈ L1 (Z̄), E f (E g (w)) ∼PX E g ◦f (w).
Proposition 3.2 and Proposition 3.3 make the following definition
well-defined.
Definition 3.4. [Functor E] A functor E : Probop → Set is defined
by:
XO
f
X̄ f−
/ E X̄
O
:=
L1 (X̄)
−
[E f (v)]∼PX
∋
O
Ef −
Ef −
Y
E
Ȳ E
/ E Ȳ
:=
L1 (Ȳ )
_
∋
[v]∼PY
We call E a conditional expectation functor .
Note that the functors L and E defined in [Adachi, 2014] from the
category χ(Ω, G) to Set are representable as L ◦ ι and E ◦ ι, respectively
by using L and E defined in this paper. That is, Prob is a more general
and richer category than χ, while keeping the conditional expectation
functor.
The following three propositions state basic properties of our conditional expectations, which are similar to those of classical conditional
expectations.
Proposition 3.5. [Linearity] Let f − : X̄ → Ȳ be a Prob-arrow. Then
for every random variables u, v ∈ L1 (Ȳ ) and α, β ∈ R, we have
(3.3)
−
−
−
E f (αu + βv) ∼PX αE f (u) + βE f (v).
Proposition 3.6. [Positivity] Let f − : X̄ → Ȳ be a Prob-arrow. If
a random variable v ∈ L1 (Ȳ ) is PY -almost surely positive, i.e. v ≥
−
0 (PY -a.s.), then E f (v) ≥ 0 (PX -a.s.).
Proposition 3.7. [Monotone Convergence] Let f − : X̄ → Ȳ be a
Prob-arrow, v, vn ∈ L1 (Ȳ ) be random variables for n ∈ N. If 0 ≤ vn ↑
−
−
v (PY -a.s.), then 0 ≤ E f (vn ) ↑ E f (v) (PX -a.s.).
Definition 3.8. [Unconditional Expectation] For v ∈ L1 (Ȳ ), we call
−
E !Y (v) a unconditional expectation of v, where !−
Y is a unique arrow
:
0
→
Ȳ
defined
in
Proposition
2.5.
!−
Y
6
T. ADACHI AND Y. RYU
Proposition 3.9. Let !−
Y : 0 → Ȳ be a unique Prob-arrow and v ∈
1
L (Ȳ ). Then, we have
−
E !Y (v)(∗) = EPY [v].
(3.4)
Proof.
!−
Y
∫
E (v)(∗) =
{∗}
∫
∫
!−
Y
E (v) dP0 =
!−1
Y ({∗})
v dPY = EPY [v].
v dPY =
Y
□
Proposition 3.9 asserts that our unconditional expectation is a natural extension of the classical one.
4. f − -measurability
Definition 4.1. [f − -measurability] Let f − : X̄ → Ȳ be a Prob-arrow
and v ∈ L∞ (Ȳ ). v is called f − -measurable if there exists w ∈ L∞ (X̄)
such that v ∼PY w ◦ f .
The following proposition allows us to say that an element of LȲ is
f − -measurable.
Proposition 4.2. Let f − : X̄ → Ȳ be a Prob-arrow and v1 and v2
be two elements of L∞ (Ȳ ) satisfying v1 ∼PY v2 . Then, if v1 is f measurable, so is v2 .
The next proposition is well-known. For example, see Page 206 of
[Williams, 1991].
Proposition 4.3. Let f − : X̄ → Ȳ be a Prob-arrow and v ∈ L∞ (Ȳ ).
Then, v is f − -measurable if and only if v is f −1 (ΣX )/B(R)-measurable.
Proposition 4.3 says that f − -measurability is an extension of the
classical measurability.
Theorem 4.4. Let f − : X̄ → Ȳ be a Prob-arrow , u be an element
of L1 (Ȳ ) and v be a random variable in L∞ (Ȳ ), and assume that v is
f − -measurable. Then we have
(4.1)
−
−
E f (v · u) ∼PX w · E f (u),
where w ∈ L∞ (X̄) is a random variable satisfying v ∼PY w ◦ f .
Theorem 4.4 is a generalization of a classical formula
EP [v · u | G] ∼P v · EP [u | G]
for a G-measurable random variable v.
The following theorem has some categorical taste.
A CATEGORY OF PROBABILITY SPACES
7
Theorem 4.5. Let E ⊠ L, EP1 : Probop × Prob → Set be two parallel
bifunctors defined by
E ⊠ L := □ ◦ (E × L) and EP1 := E ◦ P1
where P1 : Probop × Prob → Probop is the projection for the first
component, and □ : Set × Set → Set is a functor which sending an
ordered pair of sets to the set product of its components.
Now, for each Prob-object X̄, define a function αX̄ : L1 (X̄) ×
L∞ (X̄) → L1 (X̄) by αX̄ (⟨[u]∼PX , [v]∼PX ⟩) = [u · v]∼PX . Then the following diagram commutes.
Probop × Prob
⟨X̄, X̄⟩
O
⟨f − , Id−
X⟩
/ Set
L1 (X̄) × L∞ (X̄)
Set o
αX̄
O
⟨Ȳ , Ȳ ⟩
−
EId−
Y ×Lf
L1 (Ȳ ) × L∞ (Ȳ )
EId−
X
L1 (Ȳ ) × L∞ (X̄)
EP1
/ L1 (X̄)
Ef − ×LId−
X
⟨Ȳ , X̄⟩
−
⟨Id−
Y ,f ⟩
E⊠L
L1 (X̄)
O
Ef −
αȲ
/ L1 (Ȳ )
Probop × Prob
⟨X̄, X̄⟩
−
⟨Id−
X, f ⟩
⟨X̄, Ȳ ⟩
O
⟨f − , Id−
Y⟩
⟨Ȳ , Ȳ ⟩
In other words, α : E ⊠ L −
→ EP1 is a dinatural transformation.
5. f − -independence
Definition 5.1. [Category mpProb] A Prob-arrow f − : X̄ → Ȳ
is called measure-preserving if PY ◦ f −1 = PX . A subcategory
mpProb of Prob is a category whose objects are same as those of
Prob but arrows are limited to all measure-preserving arrows.
Franz defines stochastic independence in the opposite category of
mpProb as an example of his introducing notion of stochastic independence in monoidal categories.
Definition 5.2. [Franz, 2003] Two mpProb-arrows f − : X̄ → Z̄ and
g − : Ȳ → Z̄ are called independent if there exists an mpProb-arrow
q − : X̄ ⊗ Ȳ → Z̄ such that the following diagram commutes
Z̄
w; O cGGG g−
GG
wwq−
w
GG
w
ww
GG
w
w
o
/
X̄ − X̄ ⊗ Ȳ − Ȳ
f−
p1
p2
where X̄ ⊗ Ȳ := (X × Y, ΣX ⊗ ΣY , PX ⊗ PY ), p1 and p2 are projections,
ΣX ⊗ ΣY is the smallest σ-algebra of X × Y making both p1 and p2
measurable, and PX ⊗PY is a product measure such that (PX ⊗PY )(A×
B) = PX (A)PY (B) for all A ∈ ΣX and B ∈ ΣY .
8
T. ADACHI AND Y. RYU
Franz shows that the notion of independence defined in Definition
5.2 exactly matches the classical one in the sense of the following proposition.
Proposition 5.3. [Franz, 2003] Two mpProb-arrows f − : X̄ → Z̄
and g − : Ȳ → Z̄ are independent if and only if for every pair of A ∈ ΣX
and B ∈ ΣY
(5.1)
PZ (f −1 (A) ∩ g −1 (B)) = PZ (f −1 (A))PZ (g −1 (B)).
Before extending the notion of independence to the category Prob,
we need the following note.
Proposition 5.4. Let f − : X̄ → Ȳ be a Prob-arrow. We define a
Prob-object X̄f − by
X̄f − := (X, ΣX , PY ◦ f −1 ).
(5.2)
Then, the following diagram commutes in Prob
X̄
id−
X
f−
/ Ȳ
|>
|
|
||
|| f ∼
X̄f −
where f ∼ and id−
X are corresponding Prob-arrows of f : Ȳ → X̄f − and
idX : X̄f − → X̄, respectively. Moreover, f ∼ is measure-preserving.
Definition 5.5. [Independence in Prob] Two Prob-arrows f − : X̄ →
Z̄ and g − : Ȳ → Z̄ are called independent if there exists a measurepreserving arrow q − : X̄f − ⊗ Ȳg− → Z̄ such that the following diagram
commutes.
X̄
id−
X
X̄f −
f−
g−
o
/
Ȳ
r9 Z̄O eKKKK ∼
f ∼ rrr
g
K
K
KKK
id−
rr q−
Y
rr
K
r
K
r
/ X̄f − ⊗ Ȳg− o
Ȳg−
p−
1
p−
2
Lemma 5.6. For a measure-preserving Prob-arrow f − : X̄ → Ȳ ,
Ef − ◦ Lf − = idLX̄ .
Lemma 5.7. Let X̄ and Ȳ be probability spaces. Then for all v ∈
L∞ (Ȳ ),
−
E p1 (v ◦ p2 ) ∼PX EPY [v]1X
(5.3)
where p1 and p2 are projections from X × Y to X and Y , respectively.
Theorem 5.8. Let f − : X̄ → Z̄ and g − : Ȳ → Z̄ be two independent
Prob-arrows. Then, for every v ∈ L∞ (Ȳ ), we have
(5.4)
−
−
E f (v ◦ g) ∼PX EPZ [v ◦ g]E f (1Z ).
A CATEGORY OF PROBABILITY SPACES
9
Proof. For the diagram in Definition 5.5, apply E to its left box, and
apply L to its right box. Then, we get the following diagram.
L1O X̄ o
Eid−
X
L1 X̄f − o
Ef −
nn
nnn
n
n
n
w nn
Ef ∼
Ep−
1
Lg −
? _ L∞ Z̄ o
L∞ Ȳ
O hPPP
∼
PPLg
PPP
Lid−
Lq −
Y
PPP
? _ L∞ (X̄f − ⊗ Ȳg− ) o
L∞ Ȳg−
L1 Z̄ o
n
Eq −
L1 (X̄f − ⊗ Ȳg− ) o
Lp−
2
The left and right boxes in the above diagram commute since they are
images of functors E and L, the center box also commutes by Lemma
5.6, and so does the whole diagram. Now for v ∈ L∞ (Ȳ ), let us see the
values at the upper-left corner of the diagram developed through two
paths, which should coincide.
−
(Ef − ◦ Lg − )[v]∼PY = [E f (v ◦ g)]∼PX ,
−
−
−
−
−
idX
(Eid−
◦ E p1 )(v ◦ p2 )]∼PX .
X ◦ Ep1 ◦ Lp2 ◦ LidY )[v]∼PY = [(E
Hence by Lemma 5.7,
−
−
−1
−
E f (v ◦ g) ∼PX EPZ ◦g [v]E idX (1X ) ∼PX EPZ [v ◦ g]E idX (1X ).
Now for every A ∈ ΣX ,
∫
∫
id−
X
E (1X ) dPX =
1X d(PZ ◦ f −1 ) = (PZ ◦ f −1 )(A)
A
∫A
∫
−
=
1Z dPZ =
E f (1Z ) dPX .
f −1 (A)
Therefore, E
id−
X
−
A
(1X ) ∼PX E f (1Z ), which completes the proof.
□
Definition 5.9. [f − -independence] For a random variable v ∈ L1 (Ȳ ),
we define a probability space R̄v by R̄v := (R, B(R), PY ◦ v −1 ). Then,
v − : R̄v → Ȳ is a Prob-arrow. Now for a Prob-arrow f − : X̄ → Ȳ , v
is said to be independent of f − , denoted by v⊥f − , if f − and v − are
independent.
The following proposition allows us to say that an element of L1 (Ȳ )
is independent of f − .
Proposition 5.10. Let f − : X̄ → Ȳ be a Prob-arrow and v1 and v2
be two elements of L1 (Ȳ ) satisfying v1 ∼PY v2 . Then, v1 ⊥f − implies
v2 ⊥f − .
Proof. Assume that v1 ∼PY v2 and v1 ⊥f − . Let N := {y ∈ Y | v1 (y) ̸=
v2 (y)} and M := Y − N . Then, PY (N ) = 0.
First, we show that
(5.5)
PY ◦ v1−1 = PY ◦ v2−1 .
For every B ∈ B(R) and i = 1, 2,
(PY ◦vi−1 )(B) = PY (vi−1 (B)∩N )+PY (vi−1 (B)∩M ) = PY (vi−1 (B)∩M ).
10
T. ADACHI AND Y. RYU
But,
y ∈ v1−1 (B) ∩ M ⇔ v1 (y) = v2 (y) ∈ B ⇔ y ∈ v2−1 (B) ∩ M,
which proves (5.5). Hence, R̄v1 = R̄v2 .
Now since v1 ⊥f − , we have the following measure-preserving q1− .
f−
X̄
id−
X
/
r9 ȲO eKKK v−
r
r
KK 1
KK
rrr q1−
r
KK
r
K
rr
/ X̄f − ⊗ R̄v o
R̄v
f∼
X̄f −
p−
1
1
p−
2
1
Then, q1 should satisfy that q1 (y) = ⟨f (y), v1 (y)⟩ for all y ∈ Y . Similarly we define a function q2 : Y → X × R by q2 (y) = ⟨f (y), v2 (y)⟩ for
all y ∈ Y . Then, all we need to show is that q2− is a measure-preserving
Prob-arrow, in other words,
(5.6)
PY ◦ q2−1 = (PY ◦ f −1 ) ⊗ (PY ◦ v2−1 ).
However, by (5.5) and the fact that q1− is measure-preserving, it is
enough to show that
PY ◦ q1−1 = PY ◦ q2−1 .
(5.7)
For any E ∈ ΣX ⊗ B(R) and i = 1, 2,
(PY ◦ qi−1 )(E) = PY (qi−1 (E) ∩ N ) + PY (qi−1 (E) ∩ M ) = PY (qi−1 (E) ∩ M ).
But,
y ∈ q1−1 (E) ∩ M ⇔ (⟨f (y), v1 (y)⟩ ∈ E ∧ v1 (y) = v2 (y))
⇔ (⟨f (y), v2 (y)⟩ ∈ E ∧ v1 (y) = v2 (y)) ⇔ y ∈ q2−1 (E) ∩ M,
□
which proves (5.7).
Proposition 5.11. Let !−
Y : 0 → Ȳ be a unique Prob-arrow and
1
v ∈ L (Ȳ ). Then, v is independent of !−
Y.
□
Proof. Obvious.
Theorem 5.12. Let f − : X̄ → Ȳ be a Prob-arrow and v ∈ L1 (Ȳ )
which is independent of f − . Then we have,
(5.8)
−
−
E f (v) ∼PX EPY [v]E f (1Y ).
As a combination of (5.8) and (3.4), we have
(5.9)
−
−
−
E f (v) ∼PX E !Y (v)(∗)E f (1Y ),
which is a natural generalization of the relationship between classical
conditional expectations given independent σ-fields and unconditional
expectations.
A CATEGORY OF PROBABILITY SPACES
11
6. Monetary Value Measures
A monetary value measure is defined as a presheaf on Prob.
In the following, for u1 , u2 ∈ L1 (X̄), we write u1 ≲PX u2 when
u1 ≤ u2 PX -a.s..
monetary value
Definition 6.1. [Monetary Value Measures] A
measure is a contravariant functor
Φ : Probop → Set
defined by
X̄ XO
f
Φ
f−
:=
L1 (X̄)
−
[φf (v)]∼PX
∋
O
Φf −
Φf −
Ȳ Y
/ ΦX̄
O
Φ
/ ΦȲ
:=
L1 (Ȳ )
_
∋
[v]∼PY
−
where φf satisfies
−
(1) Cash invariance: (∀v ∈ L∞ (Ȳ ))(∀u ∈ L∞ (X̄)) φf (v + u ◦
−
f ) ∼PX φf (v) + u,
(2) Monotonicity: (∀v1 ∈ L∞ (Ȳ ))(∀v2 ∈ L∞ (Ȳ )) v1 ≲PY v2 ⇒
−
−
φf (v1 ) ≲PY φf (v2 ),
−
(3) Normalization: φf (0Y ) ∼PX 0X if f − is measure-preserving,
−
(4) v ∈ L∞ (Ȳ ) implies φf (v) ∈ L∞ (X̄) if f − is measure-preserving.
We sometimes write Φ[φ· ] for Φ for explicitly noting that arrows mapped
by Φ are determined by φ· .
The most crucial point of Definition 6.1 is that φ does not move only
in the direction of time but also moves over several absolutely continuous probability measures internally. This means we have a possibility
to develop risk measures including ambiguity within this formulation.
Another key point of Definition 6.1 is that φ is a contravariant functor. So, for any pair of arrows X̄ − / Ȳ − / Z̄ in Prob, we have
f
(6.1)
g
−
−
−
−
ΦId−
X = IdL1 (X̄) and Φf ◦ Φg = Φ(g ◦ f ).
As an example of monetary value measures, we will introduce a notion of entropic value measures that depend on conditional expectations
−
E f (v) of v along f − .
Proposition 6.2. [Entropic Value Measures] Let f − : X̄ → Ȳ be
a Prob-arrow, and λ be a positive real number. Define a function
−
φf : L1 (Ȳ ) → L1 (X̄) by
(6.2)
−
−
φf (v) := λ−1 log E f (eλv ), (∀v ∈ L1 (Ȳ )).
Then, Φ := Φ[φ· ] is a monetary value measure. We call this Φ an
entropic value measure.
12
T. ADACHI AND Y. RYU
Here are some properties of monetary value measures.
Theorem 6.3. Let Φ = Φ[φ· ] : Probop → Set be a monetary value
measure, and X̄
f−
/ Ȳ
g−
/ Z̄ be arrows in Prob.
(1) If f − is measure-preserving, we have Φf − ◦ Lf − = IdLX̄ .
(2) Idempotence: If f − is measure-preserving, we have Φf − ◦Lf − ◦
Φf − = Φf − .
[
(3) Local property:
(∀v1 ∈ LȲ )(∀v2 ∈ LȲ )(∀A ∈ ΣX ) Φf − 1f −1 (A) v1 +
]
1f −1 (Ac ) v2 ∼ = [1f −1 (A) ]∼PX Φf [v1 ]∼PY +[1f −1 (Ac ) ]∼PX Φf [v2 ]∼PY .
PY
(4) Dynamic programming principle: If g − is measure-preserving,
−
−
−
−
−
we have φg ◦f (w) = φg ◦f (φg (w) ◦ g) for w ∈ L∞ (Z̄).
−
(5) Time consistency: (∀w1 ∈ L∞ (Z̄))(∀w2 ∈ L∞ (Z̄)) φg (w1 ) ≲PY
−
−
−
−
−
φg (w2 ) ⇒ φg ◦f (w1 ) ≲PX φg ◦f (w2 ).
In Theorem 6.3, two properties, dynamic programming principle and
time consistency are usually introduced as axioms ([Detlefsen and Scandolo, 2006]).
But, we derive them naturally here from the fact that the monetary
value measure is a contravariant functor.
References
[Adachi, 2014] Adachi, T. (2014). Toward categorical risk measure theory. Theory
and Applications of Categories, 29(14):389–405.
[Adachi and Ryu, 2016] Adachi, T. and Ryu, Y. (2016). A category of probability
spaces. arXiv:1611.03630 [math.PR].
[Adachi and Ryu, 2017] Adachi, T. and Ryu, Y. (2017). Monetary value measures
in a category of probability spaces. arXiv:1702.01175 [q-fin.MF].
[Detlefsen and Scandolo, 2006] Detlefsen, K. and Scandolo, G. (2006). Conditional
and dynamic convex risk measures. Working paper.
[Franz, 2003] Franz, U. (2003). What is stochastic independence? In Obata, N.,
Matsui, T., and Hora, A., editors, Quantum probability and White Noise Analysis, Non-commutativity, Infinite-dimensionality, and Probability at the Crossroads,
QP-PQ, XVI, pages 254–274, Singapore. World Sci. Publishing.
[Giry, 1982] Giry, M. (1982). A categorical approach to probability theory. In Banaschewski, B., editor, Categorical Aspects of Topology and Analysis, volume 915
of Lecture Notes in Mathematics, pages 68–85. Springer-Verlag.
[Lawvere, 1962] Lawvere, F. W. (1962). The category of probabilistic mappings.
Preprint.
[MacLane, 1997] MacLane, S. (1997). Categories for the Working Mathematician.
Number 5 in Graduate Texts in Mathematics. Springer-Verlag, New York, 2nd
edition.
[Williams, 1991] Williams, D. (1991). Probability with Martingales. Cambridge University Press.
BKC Research Organization of Social Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577 Japan
E-mail address: Takanori Adachi <[email protected]>
A CATEGORY OF PROBABILITY SPACES
13
Department of Mathematical Sciences, Ritsumeikan University, 11-1 Nojihigashi, Kusatsu, Shiga, 525-8577 Japan
E-mail address: Yoshihiro Ryu <[email protected]>