Discussion Paper No.481

On the multiplicative law ofsubjective probability
by Mitsunobu MIYAKEMarch 22, 2024

TOHOKU ECONOMICS RESEARCH
GROUPDiscussion Paper

GRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANOn the multiplicative law of subjective probability

by Mitsunobu MIYAKE*TERG Discussion Paper No.481

Graduate School of Economics and Management
Tohoku University, Sendai 980-8576, Japan

March 2024

Abstract: This note elaborates on Luce and Narens' (1978) axiomatic derivation ofsubjective probability, which generically satisfies the multiplicative law by removing
requirement for the derived subjective probability to be non-atomic. For the two original
sample spaces, we add the sample space defined by the direct product of the original sample
spaces and the sample space of the auxiliary experiment. As a main result, the necessary
and sufficient conditions are provided for the likelihood relation on the events of thesample spaces to be represented by the subjective probability satisfying the law withrespect to the direct product, allowing atoms in the original sample spaces.

Key words: subjective probability, multiplicative law, likelihood relation, auxiliary
experiment*This work is supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for

Scientific Research (C) 23K01301.

1

1. Introduction
The subjective probability theory attempts to derive a probability measure as a (normalized) numerical indicator representing a decision-maker's likelihood relation on the sample
space.1 In particular, assuming that the sample space is given by the direct product of two
sample spaces, Luce and Narens (1978) axiomatically derived the subjective probability
measures, which generically satisfy the multiplicative law with respect to the direct
product; however, the subjective probability measures are derived within the class of nonatomic measures. Because all probability measures in a finite sample space will never be
non-atomic, the requirement for being a non-atomic measure is restrictive. This note
attempts to elaborate on Luce and Narens' (1978, Theorem 5) axiomatic derivation, without
requiring the derived subjective probability to be non-atomic.2
In the decision-maker's environment, we assume the presence of two original
experiments (Experiments 1 and 2) initially. We then construct a subjctive probability
measure on each of the sample spaces corresponding to the two experiments. To consider
the multiplicative law between the two subjctive probability measures, we introduce thethird experiment (Experiment 3) in which sample space is defined by the direct product of
the two sample spaces. As the fourth experiment (Experiment 4), the auxiliary experiment
by DeGroot (1970, Section 6.3) and French (1982, 2022) is introduced to determine thesubjective probability of an event in the other sample spaces above based on the relative
size of the equally likely event in the auxiliary experiment.

1

For the survey articles, see Fishburn (1986, 1994).

2

In Savage's scheme, where both the subjective probabilityand the expected utility are derived fromthe preference ordering, Grabisch, Monet and Vergopoulos (2023, Theorem 7) derive the subjective
probability satisfying the multiplicative law in case of the non-atomic subjective probability.

2First, we introduce French's (1982, 2022) five conditions for the likelihood relationon the events of the four sample spaces, assuming that the decision-maker's likelihood
comparisons are possible for any pair of events, even if the events are selected from thedifferent sample spaces. As a consequence of French (1982, 2022 Lemma 1), we show thatthe five conditions are necessary and sufficient for the likelihood relation to be
represented by subjective probability, allowing atoms in the original sample spaces(Theorem 1).
Second, the domain of the likelihood relation is extended by including all of theconditional evens in the sample spaces. This extension is required for defining a condition
called "Independence of irrelevant events" (IIE), which is a key condition for the main
theorem stated below.3 We assume that the decision-maker's likelihood comparisons are
possible for any pair of conditional events, even if the conditioning events are different in
the pair. Under the five conditions for Theorem 1, the assumption of three additional
conditions is shown to be necessary and sufficient for the likelihood relation on theextended domain to be represented by the conditional probability, and that each of thethree additional conditions is indispensable for the axiomatic derivation of the conditional
probability (Theorem 2).
As the main result, under the eight conditions for Theorems 1 and 2, the assumptionof two additional conditions, including the IIE condition on the likelihood relation, is isshown to be necessary and sufficient for the subjective probability to satisfy themultiplicative law, and that each of the two additional conditions is indispensable for the3

Based on the condition that A ~ A|B in Luce and Narens (1978, Theorem 3), we define the IIE

condition in terms of the likelihood relation. The IIE condition is closely related to the stochastic
independence condition in the subjective expected utility theory such as Grabisch, Monet andVergopoulos (2023, Axiom A7).

3

axiomatic derivation (Theorem 3).

Hence, the subjective probabilities satisfying themultiplicative law are axiomatically derived from the full class of finite sample spaces,
including the repeated coin-tossing experiments.
Within the class of coin-tossing experiments, we construct an example where
Theorem 3 does not hold if the IIE condition is dropped in Theorem 3 to prove theindispensability of the IIE condition for Theorem 3. Specifically, in the case of the cointossings where the decision-maker will be informed of the outcome of the first tossing before
the second tossing, the IIE condition requires that the likelihood of an outcome of the second
tossing is invariant, whichever outcome occurs at the first tossing.

If the initial belief(likelihood) before the first tossing is firm, then the belief may not be modified, and the IIE
condition holds for such a decision-maker.
In the next section, we introduce the sample spaces and the likelihood relation on
the events of the sample spaces.

2. Sample spaces and likelihood relation
Suppose that two original experiments exist, Experiments 1 and 2. For i = 1, 2, let Si be
the set of possible outcomes, called the sample space of Experiment i. Let 𝔅i be a s-field of
subsets of Si, that is, 𝔅i is a set of subsets of Si that is closed under complementation ands-additivity. A set in 𝔅i is called an event in Si. Specifically, Si is called the total event of
𝔅i. For the purpose of convenience, the empty set in 𝔅i is denoted by ∅i. For a given pairof events A and B in 𝔅i, a conditional event is denoted by an ordered pair A|B, where A|B
represents an event A conditioned on an event B. The set of all conditional events on Si isdefined by Gi = { A|B : A ∈ 𝔅i and B ∈ 𝔅i. }.
To consider the independence between Experiments 1 and 2, we introduce the third
experiment (Experiment 3) in which sample space S is defined by the direct product, S =4

S1⨉S2 = { (s1, s2) : s1 ∈ S1 and s2 ∈ S2 }, and denote 𝔅S = 𝔅1⨉𝔅2.4 A set in 𝔅S is called an
event in S. Specifically, S is called the total event of 𝔅S, and the empty set in 𝔅S is denotedby ∅S. For a given pair of events A and B in 𝔅S, a conditional event is denoted by an orderedpair A|B, where A|B represents an event A conditioned on an event B. The set of allconditional events on S is defined by GS = { A|B : A ∈ 𝔅S and B ∈ 𝔅S }.
The fourth experiment, Experiment 4, is introduced as French's auxiliary
experiment in which sample space is the unit interval T º [0, 1]. Let 𝔅T be the set of all
Borel subsets in T. A set in 𝔅T is called an event in T. Specifically, T is called the total
event of 𝔅T, the empty set in 𝔅T is denoted by ∅T, and the set of intervals in T is denoted by
𝔅T*. We assume that { a } º [a, a] ∈ 𝔅T* for all a ∈ T, and we introduce the following two
transformations for the events in 𝔅T*:
Translation to the right: Define A+c by A+c = { x ∈ T : x = y + c for some y ∈ A } for all A ∈𝔅T* with sup A < 1 and all c ∈ (0, 1 – sup A].
Proportional shrink: Define c·A by c·A = { x ∈ T : x = c·y for some y ∈ A } for all A ∈ 𝔅T* andall c ∈ (0, 1).
In the case of sample space T, a conditional event is denoted by an ordered pair A|B for A,
B ∈ 𝔅T. The set of all conditional events on T is defined by GT = { A|B : A ∈ 𝔅T and B ∈ 𝔅T }.
A likelihood relation is a complete and transitive binary relation ≿ on its domain 𝒟⊂ G1⋃G2⋃GS⋃GT.

A likelihood relation is denoted by (𝒟, ≿).

For A|B, C|D ∈ 𝒟, theexpression A|B ≿ C|D means that the likelihood of A conditioned on B is greater than thelikelihood of C conditioned on D. The symmetric and asymmetric parts of ≿ are denoted by~ and ≻, respectively. We assume that
4

𝔅S is the minimal s-field in the class of all s-fields of the subsets of S1⨉S2 containing all the productsets A1⨉A2, where Ai ∈ 𝔅i for i = 1, 2. See Itô (1984, Ch.2, Exercise 2.1(viii)) for the definition.

5𝒟 = { A|B ∈ G1⋃G2⋃GS⋃GT : B ≻ ∅T }.
Moreover, to simplify the arguments, we introduce a definition and an assumption: let h bea function on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T defined byh(A) = S1 if A ∈ 𝔅1;= S2 if A ∈ 𝔅2;= S if A ∈ 𝔅S;= T if A ∈ 𝔅T.
Let A be an event in 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T.

Because the conditional event A|h(A) can berecognized as the event A itself, we assume that
A = A|h(A) for all A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T,
and we use the expression "A ≿ B" instead of "A|h(A) ≿ B|h(B)" when A|h(A), B|h(B) ∈ 𝒟
in the remainder of this paper.5

3. French's theorem for the existence of subjective probability
To ensure the existence of the subjective probability, French (1982, 2022 Lemma 1) provides
the necessary and sufficient conditions for the likelihood relation to be represented by
the subjective probability.

Concretely, we call the conditions axioms and we introduce

French's axioms for the likelihood relation (𝒟, ≿) as follows:
A1(Total events and empty events): S1 ~ S2 ~ S ~ T ≻ ∅1 ~ ∅2 ~ ∅S ~ ∅T and T ≿ A ≿ ∅T for
all A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T .
5

The set G1⋃G2 can be embedded into G3 by the embedding rule that A⨉S2|S1⨉S2 = A|S1 for all A ∈𝔅1 and S1⨉B|S1⨉S2 = B|S2 for all B ∈ 𝔅2. Because the embedding rule requires that S1 = S1|S1 =S1⨉S2|S1⨉S2 = S2|S2 = S2, we do not embed G1⋃G2 into G3.

6

Because h(A) ≻ ∅T for all A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T under A1, we have the following lemma:
Lemma 1: 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T ⊂ 𝒟, whenever the likelihood relation (𝒟, ≿) satisfies A1.
The next axiom is the additivity:
A2(Additivity): For any A, B, C, D ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that h(A) = h(B), h(C) = h(D), A⋂B= ∅h(A) and C⋂D = ∅h(C), (A ≿ C, B ≿ D) ⇒ A⋃B ≿ C⋃D holds, and (A ≻ C, B ≿ D) ⇒ A⋃B ≻C⋃D holds.
A3(Continuity): Let { Bn } be a sequence of events in 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T. If Bn+1 ⊂ Bn for all n= 1, 2, ⋯ , and if there exists A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that Bn ≿ A for all n = 1, 2, ⋯, then⋂nBn ≿ A, where ∩n Bn ≡ ⋂∞n=1 Bn .
A4(Positivity in 𝔅T*): sup A > inf A ⇔ A ≻ ∅T for all A ∈ 𝔅T*.
A5(Invariance against translations to the right in 𝔅T*): A ~ (A+c) for all A ∈ 𝔅T* with
sup A < 1 and all c ∈ (0, 1 – sup A].
The three axioms, A1−A3 are assumed in French (1982, 2022), and this paper introduces
A4 and A5, each of which is weaker than SP2 in French (1982) and PSW2 in French (2022).6
We have the following theorem:
Theorem 1 (French 1982 Theorem, French 2022 Lemma 1b): (A) If a likelihood relation (𝒟,≿) satisfies the axioms A1−A5, then a real-valued function p uniquely exists on
𝔅1⋃𝔅2⋃𝔅S⋃𝔅T with the following five properties:
P1(Probability on 𝔅1): The restriction of p on 𝔅1 coincides with a probability measure on S1.
P2(Probability on 𝔅2): The restriction of p on 𝔅2 coincides with a probability measure on S2.
P3(Probability on 𝔅S): The restriction of p on 𝔅S coincides with a probability measure on S.
P4(Uniform distribution on T): The restriction of p on 𝔅T coincides with the Lebesgue
6

The two axioms, SP2 in French (1982) and PSW2 in French(2022) hold under A1-A5, as shown by

Lemma 2(v) of this paper. Moreover, SP5 in French (1982) and PSW6 in French(2022) hold underA1-A5, as shown by Lemma 2(viii, ix).

7

measure µ on T.
P5(Preservation of ordering on events): p(A) ≥ p(B) ⇔ A ≿ B for all A, B ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T.
(B) If a real-valued function p* exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having the properties P1−P4, thenthe binary relation ≿* defined on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T by A ≿* B ⇔ p*(A) ≥ p*(B) for all A, B ∈𝔅1⋃𝔅2⋃𝔅S⋃𝔅T satisfies A1−A5.
Theorem 1 implies that the five axioms A1−A5 are necessary and sufficient for the
likelihood relation on the unconditional events to be represented by the probability measure.
Although Theorem 1 has been shown by French (1982), the proof of Theorem 1 is provided
in Section 6 for the completeness of this paper.

4. A theorem for the subjective conditional probability
For a given likelihood relation (𝒟, ≿) satisfying the axioms A1−A5, it holds by Theorem 1(A)that a real-valued function p exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having the properties P1−P5. We
call the function p the probability function of (𝒟, ≿). This section considers when theprobability function also satisfies the following condition:
P6(Preservation of ordering on conditional events): p(A⋂B)/p(B) ≥ p(C⋂D)/p(D) ⇔ A|B ≿C|D for all A|B, C|D ∈ 𝒟.7
Let us consider an example of the sample spaces and likelihood relation:
Example 1: Set S1 = { a, b }, S2 = { c, d }, and S = { (a, c), (a, d), (b, c), (b, d) }. Let p1 be areal-valued function defined on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that: p1(a) = p1(b) = p1(c) = p1(d) = 1/2 ;p1(a, c) = p1(a, d) = p1(b, c) = p1(b, d) = 1/4, and that the restriction of p1 on 𝔅T coincides with

7

Luce (1968, Theorem 1) derives the subjective probability measure having property P6, requiringthat the range of the derived probability measure is {0, 1}, {0, 1/2, 1}, or includes all rationales in [0,1]. This section elaborates on Luce (1968, Theorem 1), dispensing with the condition for the range.

8the Lebesgue measure on T. Denote 𝒟1 ≡ { A|B ∈ G1⋃G2⋃GS⋃GT : p1(B) > 0}, and define areal-valued function g1 on 𝒟1 by
g1(A|B) = 1/8= p1(A⋂B)/p1(B)

if A|B ∈ GS and B ≠ S
otherwise.

Define a binary relation ≿1 on 𝒟1 by
A|B ≿1 C|D ⇔ g1(A|B) ≥ g1(C|D) for all A|B, C|D ∈ 𝒟1.
Because g1(A|h(A)) = p1(A) for all A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T, it holds by Theorem 1(B) that (𝒟1, ≿1)satisfies the axioms A1−A5 and p1 has the properties P1−P5, which implies that p1 is a
probability function of (𝒟1, ≿1). Setting A* = { (a, c), (a, d), (b, c) }, it holds that
{ (a, c) }|A* ~1 {(a, c), (a, d)}|A* andp1({(a, c), (a, d)}⋂A*)/p1(A*) = [p1(a, c) + p1(a, d)]/p1(A*) = 2/3> p1({(a, c)}⋂A*)/p1(A*) = p1(a, c)/p1(A*) = 1/3,
which means that p1 does not have the property P6.
To exclude such a likelihood relation, we introduce some additional axioms for the
likelihood relation (𝒟, ≿):A6(Consistency): For all A, B, C, D ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that A ⊂ B, C ⊂ D and B ~ D ≻∅T , A ≻ C ⇒ A|B ≻ C|D holds, and A ~ C ⇒ A|B ~ C|D holds.
A7(Homogeneity in GT): For all A, B ∈ 𝔅T* with A ⊂ B and all c ∈ (0, 1), if A|B ∈ GT ⋂𝒟 andc·A|c·B ∈ GT ⋂𝒟, then A|B ~ c·A|c·B.8
A8(Essentiality): A|B ~ (A⋂B)|B for all A|B ∈ 𝒟.9

8

A closely related axiom to A7 is introduced by Miyake (2016, Homogeneity) to characterize thelogarithmically homogeneous preferences, which describe the difference comparisons in ℝn.
9

The axiom A8 is introduced by Luce (1968).

9

The three axioms A6, A7, and A8 are standard. We require a definition to state the theorem:
For a given real-valued function p* on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T, a pair (𝒟*, ≿*) is called the induced
relation of p* if and only if
𝒟* = { A|B ∈ G1⋃G2⋃GS⋃GT : p*(B) > 0 } andA|B ≿* C|D ⇔ p*(A⋂B)/p*(B) ≥ p*(C⋂D)/p*(D) for all A|B, C|D ∈ 𝒟*.

(1)

Then we have the following theorem:
Theorem 2:(A) If a likelihood relation (𝒟, ≿) satisfies A1−A8, then a probability functionof (𝒟, ≿) having property P6 exists:p(A⋂B)/p(B) ≥ p(C⋂D)/p(D) ⇔ A|B ≿ C|D for all A|B, C|D ∈ 𝒟.
(B) Suppose that a real-valued function p* exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having the properties
P1−P4, then the induced relation of p* determined by (1) is a likelihood relation satisfying
A1−A8. (C) Each of the three axioms A6, A7, and A8 is independent of the other axioms in
Theorem 2(A).

Theorem 2 is proved in Section 6 of this paper. Theorem 2(A, B) implies that the eightaxioms A1−A8 are necessary and sufficient for the likelihood relation on the conditional
events to be represented by the conditional probability measure. Theorem 2(C) implies thateach of the three axioms A6, A7, and A8 is indispensable for Theorem 2(A).

5. A theorem for the multiplicative law of subjective probability
For a given likelihood relation (𝒟, ≿) satisfying the axioms A1−A8, it holds by Theorem 2(A)that there exists a probability function of (𝒟, ≿) having the property P6. This section
considers when the probability function also has the following property:
P7(Multiplicative law): p(A)∙p(B) = p(A⨉B) for all A ∈ 𝔅1 and all B ∈ 𝔅2.10
10This assertion means that p(A|S1)∙p(B|S2) = p(A⨉B|S1⨉ S2) for all A ∈ 𝔅1 and all B ∈ 𝔅2.

10

Let us consider an example of the sample spaces and likelihood relation:
Example 2 (Itô, 1984, Section 1.3c): Set S1 = { H1, T1 }, S2 = { H2, T2 }, and S = { (H1, H2),
(H1, T2), (T1, H2), (T1, T2) }, and let p2 be a real-valued function defined on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T
such that p2(H1) = p2(T1) = p2(H2) = p2(T2) = 1/2 and p2(H1, H2) = p2(T1, T2) = 3/8, p2(H1, T2) =p2(T1, H2) = 1/8, where the restriction of p2 on 𝔅T coincides with the Lebesgue measure on
T. Let (𝒟2, ≿2) be the induced relation of p2. Then, p2 being a probability function of (𝒟2, ≿2)holds, and that p2 has property P6. However, p2(H1)⋅p2(H2) = 1/4 < p2(H1, H2) = 3/8 holds,
implying that p2 does not have property P7.
To exclude such a likelihood relation, we introduce some additional axioms for the
likelihood relation (𝒟, ≿):A9(Independence of irrelevant experiment): A|S1 ~ A⨉S2|S1⨉S2 holds for all A ∈ 𝔅1, and
B|S2 ~ S1⨉B|S1⨉S2 holds for all B ∈ 𝔅2.
A10(Independence of irrelevant events): Let A be an event in 𝔅2. S1⨉A|S1⨉S2 ~S1⨉A|B⨉S2 holds for all B ∈ 𝔅1 with B ≻ ∅T.11
Axiom A9 requires that the likelihood of an event A|S1 in Experiment 1 coincides
with that of the (produced) event A⨉S2|S1⨉S2 in Experiment 3, because S2 is redundant
when one specifies the event A⨉S2 conditioned on S1⨉S2.
Axiom A10 requires that the likelihood of the (producted) event S1⨉A in Experiment
3 is independent of any conditioning events in Experiment 1. Namely, A10 is the restatement of Luce and Narens' (1978, Theorem 3) condition that A ~ A|B in our setting. In
the coin-tossing experiment as in Example 2, where a coin is tossed twice, A10 requires thatA⨉S2|S1⨉S2 ~ A⨉ S2|S1⨉B holds for all A ∈ 𝔅1 and all B ∈ 𝔅2 with B ≻ ∅T under A1−A10, because
Theorem 3 holds under A1−A10 and we have that p(A) = p(A⨉B)/p(B), implying A⨉S2|S1⨉S2 ~A⨉S2|S1⨉B.
11

11

the decision-maker's likelihood of an event in the second tossing conditioned on the outcomeof the first tossing is invariant whichever outcome is drived at the first tossing. Specifically,
the following holds:
p2(H1,H2)/[p2(H1,H2)+p2(H1,T2)] = 3/4 > p2(T1,H2)/[p2(T1,H2)+p2(T1,T2)] = 1/4,

(2)

which implies that S1⨉{H2}|{H1}⨉S2 ≻2 S1⨉{H2}|{T1}⨉S2.12 Because axiom A10 requiresthat S1⨉{H2}|{H1}⨉S2 ~2 S1⨉{H2}|{T1}⨉S2, the likelihood relation (𝒟2, ≿2) does not satisfy
A10, and A10 excludes (𝒟2, ≿2).
As a main result of this paper, we have the following theorem:
Theorem 3: (A) If a likelihood relation (𝒟, ≿) satisfies all through the axioms A1−A10, thenthere exists a probability function of (𝒟, ≿) having the properties P1−P7. (B) Suppose thata real-valued function p* exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having properties P1−P4 and P7, thenthe induced relation of p* determined by (1) is a likelihood relation satisfying all throughaxioms A1−A10. (C) Each of the two axioms A9 and A10 is independent of the other axioms
in Theorem 3(A).

Theorem 3 is proved in the next section. Theorem 3(A, B) implies that the ten axioms
A1−A10 are necessary and sufficient for the likelihood relation to be represented by theprobability satisfying the multiplicative law. Theorem 3(C) implies that each of the two
axioms, A9 and A10 is indispensable for Theorem 3(A).

6. Proof of the theorems
Proof of Theorem 1: (A) Suppose that a likelihood relation (𝒟, ≿) satisfies A1−A5. We
will prove that there exists a real-valued function p on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having the properties
12This can be recognized as a result of the decision-maker's updating of the likelihood based on theoutcome of the first coin-tossing, if the decision-maker follows the maximal likelihood principle.

12

P1−P5.

We need a lemma, which is proved in Appendix:

Lemma 2: If(𝒟, ≿)satisfies A1−A5, then the following ten assertions hold: (i) { a } º[a, a] ~ ∅T for all a ∈ T. (ii) [a, b] ≻ ∅T for all a, b ∈ T with a < b. (iii) [a, b] ~ [a, b) ~ (a, b]~ (a, b) for all a, b ∈ T with a < b. (iv) [0, a] ≿ [0, b] ⇔ a ≥ b for all a, b ∈ T. (v) µ(J) ≥ µ(K)⇔ J ≿ K for all J, K ∈ 𝔅T*. (vi) A ≿ B ⇔ BC ≿ AC for all A, B ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T with h(A) =h(B), where AC ≡ { s ∈ h(A): s ∉ A }. (vii) Let { Bn } be a sequence of events in 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T.
If Bn ⊂ Bn+1 for all n, and if there exists A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that A ≿ Bn for all n, thenA ≿ ⋃nBn, where ∪n Bn ≡ ⋃%&'( Bn . (viii) For any A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T, { x ∈ T : [0, x] ≿ A } is
non-empty and closed in T. (ix) For any A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T, { x ∈ T : A ≿ [0, x] } is nonempty and closed in T. (x) For any A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T there uniquely exists a real number
x ∈ T such that A ~ [0, x].

Define a real-valued function p on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T byp(A) = x for all A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T,(3)

where x is the number determined uniquely by Lemma 2(x) for each A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T.
(p has property P5): For all A, B ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T, it holds by (3) and Lemma 2(v) that A≿ B ⇔ [0, p(A)] ≿ [0, p(B)] ⇔ p(A) ≥ p(B), which means that p has property P5.
(p has property P3): We have by (3) and A1 that p(S) = 1, and we have by (3), A1 andLemma 2(i) that p(∅S) = 0. We prove that p is finitely additive on 𝔅S. Fix any A, B ∈ 𝔅S
with A⋂B = ∅S. It holds by (3) that A ~ [0, p(A)] and B ~ [0, p(B)]. It follows from Lemma2(iii, v) that A ~ [0, p(A)] ~ [0, p(A)) and B ~ [0, p(B)] ~ [p(A), p(A)+p(B)]. We have by A2
that A⋃B ~ [0, p(A))⋃[p(A), p(A)+p(B)] = [0, p(A)+p(B)], which implies that p(A⋃B) =p(A)+p(B). Hence p is finitely additive on 𝔅S.
We prove that p is s-additive on 𝔅S.

We need a lemma proved in Appendix:

Lemma 3: If { An } is a sequence of events in 𝔅S satisfying An+1 ⊂ An for all n, and if ⋂nAn

13

= ∅S, then lim p(An) = 0.
It holds by Itô (1984, Ch.2 Theorem 2.2.4 and Remark) and Lemma 3 that p is s-additive
on 𝔅S. Thus p has property P3.
(p has properties P1 and P2): We can prove that p has properties P1 and P2, using almost
the same manner as the proof in case of P3.
(p has property P4): It holds by (3) and Lemma 2(i) that p(T) = 1 and p(∅T) = 0. Fix any
A, B ∈ 𝔅T with A⋂B = ∅. We can prove that p is finitely additive on 𝔅T by almost the samemanner in the proof above. For any intervals J ∈ 𝔅T*, we have J ~ [0, µ(J)] by Lemma 2(v),which implies p(J) = µ(J).

Hence, we have by Hopf's extension theorem (Itô, 1984, Ch.2

Theorem 2.2.4) that p(A) = µ(A) for all A ∈ 𝔅T, because µ is s-additive on 𝔅T.
(B) Suppose that there exists a real-valued function p* on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having properties
P1−P4. Let (𝒟*, ≿*) be the induced relation of p*. Then we can prove easily that (𝒟*, ≿*)satisfies all the axioms.
Proof of Theorem 2: (A) Suppose that a likelihood relation (𝒟, ≿) satisfies all the axioms.
It folllows from Theorem 1 that there exists a real-valued function p having properties P1−P5. We will prove that p has property P6. We need two lemmas proved in Appendix:
Lemma 4: (i) Fix any A2|A1, A4|A3, B2|B1, B4|B3 ∈ 𝒟. If Ai ~ Bi for i = 1, 2, 3, 4, and if
Ai+1 ⊂ Ai and Bi+1 ⊂ Bi for i = 1, 3, then A2|A1 ≿ A4|A3 ⇔ B2|B1 ≿ B4|B3. (ii) For all A,
B, C, D ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that A ⊂ B, C ⊂ D and B ~ D ≻ ∅T , C|D ≿ A|B ⇒ C ≿ A.
Lemma 5: (i) [0, b]|[0, a] ≿ [0, d]|[0, g] ⇔ b/a ≥ d/g for all a, b, g, d ∈ T such that a, g > 0, a
≥ b, g ≥ d. (ii) A|B ≿ C|D ⇔ p(A)/p(B) ≥ p(C)/p(D) for any A|B, C|D ∈ 𝒟 with A ⊂ B andC ⊂ D.
Fix any A|B, C|D ∈ 𝒟. It holds by A8 and Lemma 5(ii) that
A|B ≿ C|D ⇔ (A⋂B)|B ≿ (C⋂D)|D ⇔ p(A⋂B)/p(B) ≥ p(C⋂D)/p(D),14which menas that p has property P6.
(B) Suppose that a real-valued function p* exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having properties
P1−P4. Let (𝒟*, ≿*) be the induced relation of p*. Then we can prove easily that (𝒟*, ≿*)satisfies A1−A8.
(C) (Independency of A6): Let us consider Example 1 again. It suffices to prove that (𝒟1,
≿1) does not satisfy A6 and (𝒟1, ≿1) satisfies A7 and A8. Let p1 be the symmetric probability distribution defined in Example 1. Becausep1({(a, c), (a, d)}) = p1(a, c) + p1(a, d) = 1/2 > p1(a, c) = 1/4,
implies {(a, c), (a, d)} ≻1 {(a, c)}, it holds that{(a, c), (a, d)} ≻1 {(a, c)} and {(a, c)}|A* ~1 {(a, c), (a, d)}|A*,
which implies that (𝒟1, ≿1) does not satisfy A6. Because the restriction of p1 on 𝔅T coincides
with the Lebesgue measure on T, (𝒟1, ≿1) satisfies A7. Set G* = { A|B ∈ GS : B ≠ S }. For
A|B ∈ 𝒟1, let us consier the two cases:
Case 1(A|B ∈ G*): Because (A⋂B)|B ∈ G*, we have by the definition of ≿1 that A|B ~1(A⋂B)|B.
Case 2(A|B ∉ G*): Because (A⋂B)|B ∈ G*, we have that g1(A|B) = p1(A⋂B)/p1(B) andg1(A⋂B)|B) = p1(A⋂B)/p1(B), which implies A|B ~1 (A⋂B)|B.
Hence we have that (𝒟1, ≿1) satisfies A8.
(Independency of A7): Let us consider S1 = {a, b}, S2 = {c, d}, and S = { (a, c), (a, d), (b, c),(b, d) } again.

Let p1 be the symmetric probability distribution defined in Example 1.

Define 𝒟3 by 𝒟3 = 𝒟1, and define a real-valued function g3 on 𝒟3 by
g2(A|B) = (4/5)⋅[p1(A⋂B)/p1(B)] + (1/5)

if A|B ∈ GT and p1(A⋂B)/p1(B) > 7/12

= (1/12)⋅p1(B) + (13/24)

if A|B ∈ GT and p1(A⋂B)/p1(B) = 7/12

= (6/7)⋅[p1(A⋂B)/p1(B)]

if A|B ∈ GT and p1(A⋂B)/p1(B) < 7/12

15

= p1(A⋂B)/p1(B)

if A|B ∈ G1⋃G2⋃GS .

Define a binary relation ≿3 on 𝒟3 by A|B ≿3 C|D ⇔ g3(A|B) ≥ g3(C|D) for all A|B, C|D ∈𝒟3. Then it holds that [0, 7/24]|[0, 1/2] = 2⋅[0, 7/48]|2⋅[0, 1/4] ≻3 [0, 7/48]|[0, 1/4], whichmeans that (𝒟3, ≿3) does not satisfy A7. Moreover, it holds thatp1([0, 7/24]⋂[0, 1/2])/p1([0, 1/2]) = 7/12 = p1([0, 7/48]⋂[0, 1/4])/p1([0, 1/4]),
which means that p1 does not have the property P6 with respect to (𝒟3, ≿3), even though (𝒟3,≿3) satisfies A1−A6 and A8. Because it holds by the definition of g3(A|B) that g3(A|B) =g3((A⋂B)|B) for all A|B ∈ 𝒟3, we have that A|B ~3 (A⋂B)|B, which means that (𝒟3, ≿3)satisfies A8. For all A, B, C, D ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that A ⊂ B, C ⊂ D and B ~ D ≻ ∅T ,it holds that A ≻ C ⇒ g3(A|B) > g3(C|D), and that A ~ C ⇒ g3(A|B) = g3(C|D). Hence (𝒟3,≿3) satisfies A6.
(Independency of A8): Let us consider S1 = {a, b}, S2 = {c, d}, and S = { (a, c), (a, d), (b, c),(b, d) } again. Let p1 be the symmetric probability distribution defined in Example 1. Define
𝒟4 by 𝒟4 = 𝒟1, and efine a binary relation ≿4 on 𝒟4 by
A|B ≿4 C|D ⇔ p1(A)/p1(B) ≥ p1(C)/p1(D) for all A|B, C|D ∈ 𝒟4.
Then it holds that [0, 5/8]|[1/2, 1] ≻4 [1/2, 5/8]|[1/2, 1], which means that (𝒟4, ≿4) does notsatisfy A8. Moreover, it holds thatp1([0, 5/8]⋂[1/2, 1])/p1([1/2, 1]) = 1/4 = p1([1/2, 5/8]⋂[1/2, 1])/p1([1/2, 1]).
Hence p1 does not have property P6 with respect to (𝒟4, ≿4), even though (𝒟4, ≿4) satisfies
A1−A7.
Proof of Theorem 3: (A) Suppose that a likelihood relation (𝒟, ≿) satisfies all the axioms.
It folllows from Theorem 2 that there exists a real-valued function p having properties
P1−P6. We will prove tha p has property P7. Fix any A∈ 𝔅1 and A∈ 𝔅2.
Case 1(A ≻ ∅T): It holds by A10 that16

p(A) = p(A⨉S2) and p(B) = p(S1⨉B)(4)

It holds by A9 and Theorem 2 that p(S1⨉B) = p(S1⨉B)/p(S1⨉S2) = p(A⨉B)/p(A⨉S2). Hence
we have by (4) and this that p(B) = p(S1⨉B) = p(A⨉B)/p(A⨉S2) = p(A⨉B)/p(A), which implies
that p(A⨉B) = p(A)p(B).
Case 2(A ~ ∅T): It holds by Theorem 1 that p(A) = 0. It holds by A10 that p(A⨉S2). BecauseA⨉B ⊂ A⨉S2, we have that p(A⨉B) = 0. Hence p(A)p(B) = p(A⨉B) = 0.
(B) Suppose that a real-valued function p* exists on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T having the properties
P1−P4 and P7. Let (𝒟*, ≿*) be the induced relation of p*. Then we can prove easily that
(𝒟*, ≿*) satisfies A1−A10.
(C) (Independency of A9): Let us consider S1 = {a, b}, S2 = {c, d}, and S = { (a, c), (a, d), (b,c), (b, d) } again. Let p3 be a real-valued function defined on 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that: p3(a)= 1/4, p3(b) = 3/4; p3(c) = p3(d) = 1/2 ; p3(a, c) = p3(a, d) = p3(b, c) = p3(b, d) = 1/4, and that the
restriction of p3 on 𝔅T coincides with the Lebesgue measure µ on T. Hence it holds thatp3(a)⋅p3(S2) = 1/4 < p3({ a }⨉S2) = p3({ a }⨉{c, d})= p3({(a, c)}⋃{(a, d}) = p3(a, c) + p3(a, d) = 1/2,
which means that p3 does not have the property P7. Moreover, let (𝒟5, ≿5) be the induced
relation of p3. Namely, (𝒟5, ≿5) is determined by (1). Then it holds thatp3(a) = p3(a|S1) = 1/4 and p3({ a }⨉S2) = p3({ a }⨉S2|S1⨉S2) = 1/2,
which implies that { a }⨉S2|S1⨉S2 ≻5 a|S1. Hence (𝒟5, ≿5) does not satisfy A9. Because p3
has properties P1−P6, it holds by Theorem 2(B) that (𝒟5, ≿5) satisfies A1−A8. Moreover, itholds thatp3({ a }⨉S2) = p3(a, c) + p3(a, d) = 1/2; p3([{ a }⨉{ c }]⋂[S1⨉{ c }])/p3(S1⨉{ c })= p3(a, c)/[p3(a, c)+p3(b, c)] = (1/4)/(1/2) = 1/2,p3({ a }⨉S2) = p3(a, c) + p3(b, d) = 1/2; p3([{ a }⨉{ d }]⋂[S1⨉{ d }])/p3(S1⨉{ d })17

= p3(a, d)/[p3(a, d)+ p3(b, d)] = (1/4)/(1/2) = 1/2,p3({ b }⨉S2) = p3(b, c) + p3(b, d) = 1/2; p3([{ b }⨉{ c }]⋂[S1⨉{ c }])/p3(S1⨉{ c })= p3(b, c)/[p3(a, c) + p3(b, c)] = (1/4)/(1/2) = 1/2,p3({ b }⨉S2) = p3(b, c) + p3(b, d) = 1/2; p3([{ b }⨉{ d }]⋂[S1⨉{ d }])/p3(S1⨉{ d })= p3(b, d)/[p3(a, d) + p3(b, d)] = (1/4)/(1/2) = 1/2,
which implies that (𝒟5, ≿5) satisfies the axioms A10.
(Independency of A10): Let us consider Example 2 again. Because it follows from (2) that
(𝒟2, ≿2) does not satisfy A10, it suffices to prove that (𝒟2, ≿2) satisfies A1−A9. Let p2 be theprobability distribution defined in Example 2. Because p2 has properties P1−P6, it holds by
Theorem 2(B) that (𝒟2, ≿2) satisfies A−A8. Moreover, it holds thatp2({ H1 }⨉S2) = p2(H1, H2) + p2(H1, T2) = 1/2 = p2(H1);p2({ T1 }⨉S2) = p2(T1, H2) + p2(T1, T2) = 1/2 = p2(T1);
p2(S1⨉{ H2 }) = p2(H1, H2) + p2(T1, H2) = 1/2 = p2(H2);
p2(S1⨉{ T2 }) = p2(H1, T2) + p2(T1, T2) = 1/2 = p2(T2),
which implies that (𝒟2, ≿2) satisfies A9.

Appendix
Proof of Lemma 2: (i) It holds by A1 and A4 that { a } ~ ∅T for all a ∈ [0, 1]. (ii) Fix any
a, b ∈ [0, 1] with b > a, it holds by A4 that [a, b] ≻ ∅T. (iii) Fix any a, b ∈ [0, 1] with a <b. Because [a, b) ≿ [a, b) and { b } ≿ ∅T by A1, it holds by A2 that [a, b] ≿ [a, b). Because[a, b) ≿ [a, b) and ∅T ≿ { b } by Lemma 2(i), it holds by A2 that [a, b) ≿ [a, b]. Hence [a, b]~ [a, b). By almost the same manner we can prove that [a, b] ~ (a, b] ~ (a, b). (iv) Supposethat a ≥ b. Because [0, b) ≿ [0, b) and [b, a] ≿ ∅T by A1, it holds by A2 that [0, a] ≿ [0, b).
It holds by Lemma 2(iii) that [0, a] ≿ [0, b) ~ [0, b]. Suppose that b > a. Because [0, a) ≿[0, a) and [a, b] ≻ ∅T by Lemma 2(ii), it holds by A2 that [0, b] ≻ [0, a). It holds by Lemma18

2(iii) that [0, b] ≻ [0, a) ~ [0, a]. Hence we have that b > a ⇒ [0, b] ≻ [0, a], which implies[0, a] ≿ [0, b] ⇒ a ≥ b. (v) It follows from Lemma 2(iii) that it suffices to prove the case of
the closed intervals. For any intervals [a, b], [c, d] ∈ 𝔅T*, it holds by A5 that [0, b – a] ~[a, b] and [0, d – c] ~ [c, d].

Hence we have by Lemma 2(iv) that µ([a, b]) ≥ µ([c, d]) ⇔(b – a) ≥ (d – c) ⇔ [0, b – a] ≿ [0, d – c] ⇔ [a, b] ≿ [c, d]. (vi) Assume that A ≿ B. If AC≻ BC, it holds by A ≿ B and A2 that [AC⋃A] ≻ [BC⋃B], which is a contradiction. Thus wehave that BC ≿ AC. (vii) Suppose that Bn ⊂ Bn+1 for all n and that there exists A ∈𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such that A ≿ Bn for all n. Define Cn = BnC for all n, and define D =AC. Then it holds by Lemma 2(vi) that Cn+1 ⊂ Cn for all n and that Cn ≿ D for all n. Hence
we have by A3 that ⋂nCn ≿ D, which implies that A = DC ≿ (⋂nCn)C = ⋃nBn. (viii) Fix any
A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T. It holds by A1 that T ≿ A, which implies that 1 ∈ { x ∈ T : [0, x] ≿ A }≠ ∅T. Suppose that a sequence { xn } in T converges to x* and that [0, xn] ≿ A for all n. It
suffices to prove that x* ∈ T and [0, x*] ≿ A. Because T is closed, it holds that x* ∈ T ≠ ∅T.
Define { Bn } in 𝔅T* by Bn = [0, xn] for all n. It holds by Anderson and Hall (1963, Ch II,
Theorem 1.10 and Ch. IV, Theorem 1.9) that we can assume that xn ≥ xn+1 for all n, orthat xn ≤ xn+1 for all n, without loss of generality.
Case 1(xn ≥ xn+1 for all n): We have Bn+1 ⊂ Bn for all n by xn ≥ xn+1 for all n. Because Bn≿ A for all n, we have by A3 that ⋂nBn ≿ A. Hence it holds by ⋂nBn ⊂ [0, x*] and Lemma
2(iv) that [0, x*] ≿ A.
Case 2(xn ≤ xn+1 for all n) : We have Bn ⊂ Bn+1 for all n by xn ≤ xn+1 for all n. Because B1≿ A and B1 ⊂ ⋂nBn ⊂ [0, x*], it holds by Lemma 2(iv) that [0, x*] ≿ A.
(ix) Fix any A ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T.

It holds by A1 and Lemma 2(i) that A ≿ [0, 0], which

implies that 0 ∈ { x ∈ T : A ≿ [0, x] } ≠ ∅T. Suppose that a sequence { xn } in T converges tox* and that A ≿ [0, xn] for all n. It suffices to prove that x* ∈ T and A ≿ [0, x*]. Because19T is closed, it holds that x* ∈ T ≠ ∅T. Define { Bn } in 𝔅T* by Bn = [0, xn] for all n. It holds
by Anderson and Hall (1963, Ch II, Theorem 1.10 and Ch. IV, Theorem 1.9) that we canassume that xn ≥ xn+1 for all n, or that xn ≤ xn+1 for all n, without loss of generality.
Case 1(xn ≥ xn+1 for all n): We have Bn+1 ⊂ Bn for all n by xn ≥ xn+1 for all n. Because
A ≿ B1 and [0, x*] ⊂ B1, we have by Lemma 2(iv) that A ≿ B1 ≿ [0, x*].
Case 2(xn ≤ xn+1 for all n): We have Bn ⊂ Bn+1 for all n by xn ≤ xn+1 for all n. Because A ≿Bn for all n, it holds by Lemma 2(vii) that A ≿ ⋃nBn. Hence we have by [0, x*) ⊂ ⋃nBn andLemma 2(iv, iii) that A ≿ ⋃nBn ≿ [0, x*) ~ [0, x*].

(x) Fix any A ∈ 𝔅S⋃𝔅T.

It holds byLemma 2(viii, ix) that the two sets { x ∈ T: [0, x] ≿ A } and { x ∈ T: A ≿ [0, x] } are non-empty
and closed in T. Because it holds by Smith (1983, Ch.7, Theorem 7.4) that T is connected,
we have that { x ∈ T : [0, x] ≿ A }∩{ x ∈ T : A ≿ [0, x] } ≠ ∅T and there exists a real number x∈ T such that A ~ [0, x]. The uniqueness of x ∈ T is ensured by Lemma 2(iv).
Proof of Lemma 3: Because p is finitely additive on 𝔅S and An = An+1⋃(An/An+1) for all n,
we have that p(An) ≥ p(An+1) ≥ 0 for all n, which implies that { p(An) } is a bounded monotonesequence. Hence it holds by Smith (1983, Ch.6, Theorem 2.2) that lim p(An) exists and limp(An) ≥ 0. Suppose that lim p(An) > 0, and set L = lim p(An) > 0. It holds by (3) that An ~[0, p(An)]. Because p(An) ≥ L > 0 for all n, we have by Lemma 2(v) that An ~ [0, p(An)] ≿ [0,L] ≻ [0, 0] ~ ∅T for all n. It holds by A1 and A3 that ⋂nAn ≻ ∅T ~ ∅S. This contradicts with
⋂nAn = ∅S. Hence we have lim p(An) = 0.
Proof of Lemma 4: (i) It holds by A6 that A2|A1 ~ B2|B1 and A4|A3 ~ B4|B3. Hence wehave A2|A1 ≿ A4|A3 ⇔ B2|B1 ≿ B4|B3. (i) Fix any A, B, C, D ∈ 𝔅1⋃𝔅2⋃𝔅S⋃𝔅T such thatA ⊂ B, C ⊂ D and B ~ D ≻ ∅T. Because the contraposition of A ≻ C ⇒ A|B ≻ C|D is that
C|D ≿ A|B ⇒ C ≿ A, we have by A6 that C|D ≿ A|B ⇒ C ≿ A.
Proof of Lemma 5: (i) Case 1 (a > g): It holds by 1 > (g/a) > 0 and A7 that20

[0, b]|[0, a] ~ [0, (g/a)b]|[0, (g/a) a] = [0, (g/a)b]|[0, g].

(5)

It holds by A6 and Lemma 4(ii) that[0, (g/a)b] ≿ [0, d] ⇔ [0, (g/a)b]|[0, g] ≿ [0, d]|[0, g].

(6)

Hence we have by (5), (6) and Lemma 2(v) that[0, b]|[0, a] ≿ [0, d]|[0, g] ⇔ [0, (g/a)b]|[0, g] ≿ [0, d]|[0, g]⇔ [0, (g/a)b] ≿ [0, d] ⇔ (g/a)b ≥ d ⇔ b/a ≥ d/g.
Case 2 (a < g): It holds by 0 < a/g < 1 and A7 that[0, d]|[0, g] ~ [0, (a/g)d]|[0, (a/g)g] = [0, (a/g)d]|[0, a].
Hence we have by A6, Lemma 4(ii) and Lemma 2(v) that[0, b]|[0, a] ≿ [0, d]|[0, g] ⇔ [0, b]|[0, a] ≿ [0, (a/g)d]|[0, a]⇔ [0, b] ≿ [0, (a/g)d] ⇔ b ≥ (a/g)d ⇔ b/a ≥ d/g.
Case 3 (a = g): We have by A6, Lemma 4(ii) and Lemma 2(v) that[0, b]|[0, a] ≿ [0, d]|[0, g] ⇔ [0, b] ≿ [0, d] ⇔ b ≥ d ⇔ b/a ≥ d/g.
(ii): Fix any A|B, C|D ∈ 𝒟 with A ⊂ B and C ⊂ D. It holds by the definition of p and Lemmas
2 and 3(i) that
A|B ≿ C|D ⇔ [0, p(A)]|[0, p(B)] ≿ [0, p(C)]|[0, p(D)] ⇔ p(A)/p(B) ≥ p(C)/p(D).

References
Anderson, K.W., and Hall, D.W., 1963, "Sets, sequences and mappings", Wiley, New York.

DeGroot, M.H., 1970, "Optimal Statistical Decisions", McGraw-Hill, New York.

Fishburn, P.C., 1986, 'The axioms of subjective probability, Statistical Science 1, 345-355.

Fishburn, P.C., 1994, Utility and subjective probability, R.J. Aumann, S. Hart (Eds.),
Handbook of Game Theory Ch.39, Vol. 2, North-Holland, Amsterdam, 1397-1435.

21

French, S., 1982, On the axiomatisation of subjective probabilities, Theory and Decision 14,
19-33.

French, S., 2022, Axiomatizing the bayesian paradigm in parallel small worlds, Operations
Research, 70, 1342-1358.

Grabisch, M., Monet, B., and Vergopoulos, V., 2023, Subjective expected utility throughstochastic independence, Economic Theory 76, 723-757.

Itô, K., (1984) "Introduction to probability theory", Cambridge University Press, Cambridge.

Luce, R.D., 1968, On the numerical representation of qualitative conditional probability,
Annals of Mathematical Statistics 39, 481-491.

Luce, R. D., and Narens, L., 1978, Qualitative independence in probability theory, Theory
and Decision 9, 225 - 239.

Miyake, M., 2016, Logarithmically homogeneous preferences, Journal of Mathematical
Economics 67, 1-9.

Smith, K. T., 1983, "Primer of modern analysis", Springer-Verlag, New York.

22