NONCONGLOMERABILITY FOR COUNTABLY ADDITIVE MEASURES THAT ARE NOT κ-ADDITIVE

Abstract Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

we bypass most of this difference by exploring de Finetti/Dubins conditional probabilities associated with countably additive unconditional probabilities. Specifically, we do not require that conditional probabilities are countably additive. 2. When B is both P-null and not empty, a regular conditional probability given B is relative to a sub-σ -field A ⊆ B, where B ∈ A. But in the de Finetti/Dubins theory of conditional probability, P(·|B), depends solely on the event B and not on any sub-σfield that embeds it. Example 4.1, which we present in §4, illustrates this difference. 3. Some countably additive probabilities do not admit regular conditional distributions relative to a particular sub-σ -field, even when both σ -fields are countably generated. (See Corollary 1 in Seidenfeld, Schervish, & Kadane, 2001.) In contrast, Dubins (1975) establishes the existence of full conditional probability functions: where, given a set Ω of arbitrary cardinality, a conditional probability satisfying Definition 1.1 is defined with respect to each nonempty element of its powerset, i.e., where B is the powerset of Ω. Hereafter, we require that each probability function includes its conditional probabilities (in accord with Definition 1.1) given each nonempty event B ∈ B. However, because we investigate conditional probabilities for a countably additive unconditional probability, in light of Ulam's Theorem (1930), we do not require that B is the powerset of the state space Ω. 4. Our focus in this paper is a fourth feature that distinguishes the de Finetti/Dubins theory of conditional probability and the Kolmogorovian theory of regular conditional probability. This aspect of the difference involves conglomerability of conditional probability functions.
Let I be an index set and let π = {h i : i ∈ I} be a partition of the sure event where the conditional probabilities, P(E|h i ) are well defined for each E ∈ B and i ∈ I. DEFINITION 1.2. The conditional probabilities P(E|h i ) are conglomerable in π provided that, for each event E ∈ B and real constants k 1 and k 2 , if k 1 ≤ P(E|h i ) ≤ k 2 for each i ∈ I, then k 1 ≤ P(E) ≤ k 2 .
That is, conglomerability requires that the unconditional probability for event E, P(E) lies within the (closed) interval of conditional probability values, {P(E|h i )|i ∈ I}, with respect to elements h of a partition π .
Conglomerability is an intuitively plausible property that probabilities might be required to have. Suppose that one thinks of the conditional probability P(E|h i ) as representing one's degree of belief in E if one learns that h i is true. Then P(E|h i ) ≤ k 2 for all i in I means that one believes that, no matter which h i one observes, one will have degree of belief in E at most k 2 . Intuitively, one might think that this should imply P(E) ≤ k 2 before learning which h i is true. That is, if one knows for sure that one is going to believe that the probability of E is at most k 2 after observing which h i is true, then one should be entitled to believe that the probability of E is at most k 2 now. This paper shows that this intuition is only good when the degree of additivity of the probability matches (or exceeds) the cardinality of the partition.
Schervish, Seidenfeld, & Kadane (1984) show that if P is merely finitely additive (i.e., if P is finitely but not countably additive) with conditional probabilities that satisfy Definition 1.1, and P is defined on a σ -field of sets, then P fails conglomerability in some countable partition. That is, for each merely finitely additive probability P there is an event E, an ε > 0, and a countable partition of measurable events π = {h n : n = 1, . . . }, where The following example illustrates a failure of conglomerability for a merely finitely additive probability P in a countable partition π = {h n : n ∈ {1, 2, . . . }}, where each element of the partition is not P-null, i.e., P(h n ) > 0 for each n ∈ {1, 2, . . . }. Then, by both the theory of conditional probability according to Definition 1.1 and the theory of regular conditional distributions (ignoring the requirement that probability is countably additive), P(E | h n ) = P(E∩h n )/P(h n ) is well defined. Thus, the failure of conglomerability in this example is due to the failure of countable additivity, rather than to a difference in how conditional probability is defined. EXAMPLE 1.3 (Dubins, 1975). Let the sure event Ω = {(i, n) : i ∈ {1, 2} and n ∈ {1, 2, . . . } and B be the powerset of Ω. Let E = {{1, n} : n ∈ {1, 2, . . . }} and h n = {{1, n}, {2, n}}, and partition π = {h n : n ∈ {1, 2, . . . }}. Partially define the finitely additive probability P by: So P is merely finitely additive over E c and P(·|E c ) is purely finitely additive. It follows easily that P(h n ) = 1/2 n+1 > 0 for each n ∈ {1, 2, . . . }. Thus, P is not conglomerable in π as: Kadane, Schervish, & Seidenfeld (1996) discuss this example in connection with the value of information. Also they show (Kadane, Schervish, & Seidenfeld, 1986, Appendix) that there exist countably additive probabilities defined on the continuum such that, if conditional probabilities are required to satisfy Definition 1.1 rather than being regular conditional distributions, then nonconglomerability results in at least one uncountable partition. Here we generalize that result to κ-non-additive probabilities that are countably additive. Throughout, we assume ZFC set theory.
Let <Ω, B, P> be a measure space, with P countably additive. That is, B is a σ -field of sets over Ω. Set B is measurable means that B ∈ B. That P is a countably additive probability is formulated with either of two equivalent, familiar definitions. That these are equivalent definitions is immediate from the requirement that B is a σ -field of sets. (See, e.g., Billingsley, 1995, p. 25.) DEFINITION 1.4. Let {A i : i = 1, . . . } be a denumerable sequence of measurable, pairwise disjoint events, and let A be their union, which then is measurable as B is a σ -field. That is, A i ∩ A j = ∅ if i = j, and A = ∪ i A i . P is countably additive 1 (in the first sense) provided that P(A) = i P(A i ) for each such sequence. DEFINITION 1.5. Let {B i : i = 1, . . . } be an increasing denumerable sequence of measurable events, with B their limit, which then is measurable. That is, B i ⊆ B j if i ≤ j, and B = ∪ i B i . Then P is countably additive 2 (in the second sense) provided that P(B) = lim i P(B i ) for each such sequence. That is, P is countably additive 2 provided it is continuous over every denumerable sequence of measurable events that approximate a measurable event from below.
In this paper we examine nonconglomerability of a set of conditional probabilities {P(E|h)} that satisfy (the de Finetti/Dubins) Definition 1.1, where these conditional probabilities are associated with a countably additive unconditional probability, P, that belongs to a measure space <Ω, B, P>. How large do we require the σ -field of sets B be in order to have available sufficiently many well-defined conditional probabilities? By an important result of Ulam (1930), unless the cardinality of Ω is at least as great as some inaccessible cardinal, B cannot be the powerset of Ω. (See, e.g., Jech (1978), chap. 27.) However, without loss of generality, we may assume that the measure space is P-complete and contains each point ω ∈ Ω. That is, if N ∈ B, P(N) = 0, and E ⊆ N, then E ∈ B. See, e.g., Billingsley (1995, p. 44), Doob (1994, p. 37), or Halmos (1950.
Our principal result here asserts that, subject to several structural assumptions to assure richness of B, presented in §3.1, the nonconglomerability of P occurs in a partition by measurable events whose cardinality κ is bounded above by the extent of nonadditivity of the countably additive probability P. We postpone to the concluding §5 our discussion of the consistency of these structural assumptions.
There are two, parallel definitions for generalizing from countable additivity (also denoted σ -additivity) to κ-additivity. In the following, let α and β be ordinals, and λ and κ be cardinals. DEFINITION 1.6. Let {A α : α < λ ≤ κ} be a sequence of λ-many measurable, disjoint events, and let A be their union, which also is presumed measurable. That is, Note: The infinite sum of a sequence of non-negative terms is the supremum over all finite sums in the sequence. When the elements of the sequence are probabilities for terms in a partition, at most countably many terms are positive. DEFINITION 1.7 (Armstrong & Prikry, 1980).
That is, P is κ-additive 2 provided that probability is continuous from below over λ-long sequences of measurable events that approach a measurable event from below.
Next, we show that for a complete measure space, κ-additive 1 is sufficient for κ-additive 2 . LEMMA 1.8. Let <Ω, B, P> be a P-complete measure space with |Ω| = κ. P is κ-additive 1 only if it is κ-additive 2 .
Proof. The sufficient Condition {*} is trivially satisfied when κ = ℵ 1 . That is, since B is a σ -field, each {B α : α < λ} is measurable. ♦Corollar y 1.11 In the light of Lemma 1.8 , in order to generalize nonconglomerability to countably additive measures, we consider P-complete measure spaces that are not κ-additive 2 , and therefore not κ-additive 1 . Trivially, when P is not λ-additive 2 and λ < κ, then P is not κ-additive 2 . So, when P is not additive 2 , we focus on the least cardinal κ where P is not κ-additive 2 .
In particular, let κ be the least cardinal where P is not κ-additive 2 , and κ ≥ ℵ 1 . Then κ is a regular cardinal. This is immediate from the observation that if P fails to be κ-additive 2 on the upward nested sequence of measurable events {B α : α < κ}, with measurable limit B, then P fails to be κ-additive 2 on each cofinal subsequence of the sequence {B α }. So, as κ is the least cardinal where P is not κ-additive 2 , then κ = cofinality(κ).
Consider a P-complete measure space <Ω, B, P>, where each point ω ∈ Ω is measurable (so B is an atomic algebra), and where P is countably additive but not κ-additive 2 . Here we show the main Proposition of this paper: • Subject to several structural assumptions on B (presented in §3.1) the probability P fails to be conglomerable in some partition π of measurable events, where the cardinality of π at most κ.
Thus, rather than thinking that nonconglomerability is an anomalous feature of finite but not countably additive probabilities, and that nonconglomerability arises solely with finitely but not countably additive probabilities in countable partitions, here we argue for a different conclusion. Namely, subject to several structural assumptions, we show that the cardinality λ of a partition where P is nonconglomerable is bounded above by the (least) cardinal for which P is not κ-additive 2 (and assuming that cardinal is not weakly inaccessible). §2. Tiers of points. The proof of the main Proposition is based on the structure of a linear order over equivalence classes (which we call tiers) of points in Ω defined by the following relation between pairs of points. DEFINITION 2.1. Consider the relation, ∼, of relative-non-nullity on pairs of points in Ω. That is, for points, ω α and ω β , they bear the relation ω α ∼ ω β provided that, either ω α = ω β , or else ω α = ω β and 0 < P({ω α }|{ω α , ω β }) < 1.

Structural assumptions for the Proposition.
The Proposition asserts that, subject to the six structural assumptions on B, presented below, when P is not-κ-additive 2 (and κ is least) then nonconglomerability obtains in some partition whose cardinality is bounded above by the same cardinal, κ.
We use a familiar partition of the fine structure of linear orderings to create three cases around which the proof of the main proposition is organized: CASE 1: The linear order ↑ is a well order on the set of tiers. CASE 2: The linear order ↓ is a well order on the set of tiers. CASE 3: There are two countable subsets L ↓ = {τ 1 , . . . , τ n , . . . } and M ↑ = {τ 1 , . . . , τ n , . . . } of the set of tiers, each well ordered as the natural number (N <), respectively, by ↓ and ↑.
As explained below, the proof of the Proposition is organized using five lemmas (Lemmas 3.3-3.5, 3.7, and 3.8) in different combinations over these three cases. Moreover, regarding the six structural assumptions, these too are used in different combinations for the five different Lemmas. Thus, which subset of the six structural assumptions is used depends upon which of the three cases arises.
Consider the measure space <Ω, B, P>. Regarding the cardinality κ of P's nonadditivity 2 , we assume that κ is not a weakly inaccessible cardinal. Combining this with the fact that κ is regular (proven above), we have that the set of cardinals less than κ has cardinality less than κ-used in the proof of Lemma 3.7.
Next, we state the six structural assumptions that we impose on B in order to secure sufficiently many measurable events for proving the central proposition. We discuss the nature of these assumptions further in §5.
DEFINITION 3.1. When T is a set of tiers, denote by ∪T the subset of Ω formed by the union of elements in T, the union of the tiers in T.
(Note that if P(T) = 0, since P is complete, each subset of T is measurable.) T odd is the set of tiers with "odd" ordinal index, ending "2n-1" for a positive integer n > 0. Then ∪T odd is measurable. T even is the set of tiers with "even" ordinal index, ending "2n" for a positive integer n > 0. Then ∪T even is measurable.
(Used with Lemma 3.7.) It is immediate from SA 5 that when ↓ or ↑ is a well order of the set of tiers in T then the set of points in tiers of T with limit ordinal index, ∪T limit , also is measurable-since {T odd , T even , T limit } forms a partition of T.

The Proposition and its proof.
PROPOSITION 3.2. Let <Ω, B, P> be a P-complete, countably additive measure space with conditional probabilities satisfying Definition 1.1, and which satisfies the six Structural Assumptions of §3.1. Assume that P fails to be κ-additive 2 for a cardinal κ, that κ is the least such cardinal, and that it is not weakly inaccessible. Then, there is a partition π = {h ι : ι ∈ I} of measurable events, where | π | ≤ κ and where P fails to be conglomerable in π . That is, there exists a measurable event E, and an ε > 0 where P(E) > P(E|h) + ε for each h ∈ π . ♦Proposition 3.2 As noted above, the proof of the Proposition 3.2 proceeds using the five Lemmas 3.3-3.5, 3.7, and 3.8. Lemmas 3.3 and 3.4 provide, respectively, one of two nonexclusive, nonexhaustive Sufficient Conditions for nonconglomerability of P. That is, there are models of the linear order of tiers satisfying each of the four Boolean combinations of these two Sufficient Conditions. Sufficient Condition 1: There is a tier τ belowτ that is not null, P(τ ) > 0. Lemma 3.3 establishes that then P is nonconglomerable.
Sufficient Condition 2: There exist two sets of tiers, U and V, with P(∪V) > 0 and | ∪U| = | ∪V|, but where U is above V in the linear ordering of tiers. That is, for each tier τ 1 in U and each tier τ 2 in V, τ 1 ↓ τ 2 : Lemma 3.4 establishes then P is nonconglomerable.
Lemmas 3.5, 3.7, and 3.8 address, respectively, one of the three exclusive and mutually exhaustive Cases for the linear order of tiers, repeated here for convenience.
CASE 1: The linear order ↑ is a well order on the set of tiers. Lemma 3.5 establishes that P is nonconglomerable in this case.
CASE 2: The linear order ↓ is a well order on the set of tiers. Lemma 3.7 establishes that P is nonconglomerable in this case. CASE 3: There are two countable subsets L ↓ = {τ 1 , . . . , τ n , . . . } and M ↑ = {τ 1 , . . . , τ n , . . . } of the set of tiers, each well ordered as the natural number (N <), respectively, by ↓ and ↑. Lemma 3.8 establishes that P is nonconglomerable in this case.
The proofs of Lemmas 3.5, 3.7, and 3.8 rely on the two facts established by Lemmas 3.3 and 3.4 that, if either of the two Sufficient Conditions obtains within one of the three Cases, then P is nonconglomerable.
Proof of Proposition 3.2. Let κ be the least cardinal for which P is not κ-additive 2 . As noted before, then κ is a regular cardinal. LEMMA 3.3. Suppose there exists a non-null tier (of null points), τ =τ , P(τ ) > 0-Sufficient Condition 1-then P is not conglomerable.
Proof. By the splitting condition, SA 4 , partition τ into two disjoint measurable sets, We identify a partition with cardinality κ, which we write as π = {h α : α < κ}, where P(T 1 | h) < d/2 for each h ∈ π . Each element h ∈ π is a finite set. Each element h α contains at most one point from T 1 , and some positive finite number of points from Ω−T 1 , selected to insure that P(T 1 | h) < d/2.
In order to complete the partition π , consider a catch-all set S with all the remaining points ω β ∈ Ω − ∪ 0<β<λ h β . Note that each point ω ∈ S is not a member of T 1 . So, for each ω ∈ S, P(T 1 | {ω}) = 0. So, for each point, ω ∈ S, add {ω} as a separate partition element of π . This insures that | π | = κ and that P is not conglomerable in π as P(T 1 ) = d > 0, yet for each h ∈ π , P(T 1 | h) < d/2. ♦Lemma 3.3 In §4, with Example 4.1, we illustrate the first Sufficient Condition and the argument of Lemma 3.3 using an ordinary continuous random variable. We use Example 4.1 to explain a difference between the de Finetti/Dubins' theory of conditional probability (Definition 1.1), and the familiar theory of regular conditional distributions. Next, Lemma 3.4 establishes Sufficient Condition 2 where P is nonconglomerable in a κ-sized partition of measurable events. We use Lemma 3.4 frequently in the arguments for Lemmas 3.5, 3.7, and 3.8. LEMMA 3.4. Let each of U and V be two disjoint sets of tiers, with ∪V a measurable set. (It is not necessary that ∪U is B-measurable.) Assume |∪U|=|∪V| = λ ≤ κ, and with U entirely above V in the linear ordering of ↓ tiers. That is, for each pair τ U ∈ U and τ v ∈ V, τ U ↓τ V . If P(∪V) > 0, then P is not conglomerable.

LEMMA 3.5. Suppose that, apart fromτ , each tier in the linear order ↑ is null (otherwise apply Lemma 3.3) and that ↑ is a well order-Case 1. Then P is not conglomerable.
Proof. We index the well order ↑ of these null tiers with an initial segment of the ordinals. Let β be the least ordinal in this well order such that P(∪ α<β τ α ) > 0 and let R be this set of tiers. R = {τ α : α < β). By SA 3 , ∪R is measurable and let |∪R| = λ ≤ κ. Evidently, we may assume that β is an uncountable limit ordinal, since P(τ α ) = 0 for each tier other thanτ .
Use SA 5 to partition R into two disjoint sets of tiers, T 1 and T 2 , each with cardinality λ. For example, T 1 might be the set of tiers with successor ordinal index-the union of T odd and T even . And T 2 might be the set of tiers with limit ordinal index. Then each of T 1 and T 2 is cofinal in the well order, ↑, of R. It is then an elementary fact that there exist a pair of injective (increasing) functions f :∪T 1 → ∪T 2 and g:∪T 2 → ∪T 1 where P({ω} | {ω, f (ω)}) = 0 and P({ω} | {ω, g(ω)}) = 0, whenever ω is in the domain, respectively, of the function f or g, i.e., whenever ω ∈ ∪T 1 or ω ∈ ∪T 2 , respectively. That is, each of f and g maps each element of its domain into a distinct element of its range belonging to a higher tier in the well order ↑. In other words, f pairs each point in ∪T 1 with a point in ∪T 2 having a higher tier under ↑. Likewise, g pairs each point in ∪T 2 with a point in ∪T 1 having a higher tier under ↑.
Use the functions f and g to create two κ-size partitions, π f and π g , as defined below, and similar in kind to the partition used in Lemma 3.3. Without loss of generality, when considering f (respectively, g), index its domain-for f that is the set of points ω ∈ ∪T 1 (respectively, for g, that is the set of points ω ∈ ∪T 2 )-using an initial segment of ordinals running through λ. That is, when considering f , write ∪T 1 = {ω 1 1 , ω 1 2 , . . . , ω 1 α , . . . } with 0 < α < λ. Similarly for g. Write ∪T 2 = {ω 2 1 , ω 2 2 , . . . , ω 2 α , . . . }. For each ordinal 0 < α < λ, define the partition element h α of π f to be the pair-set h α = {ω 1 α , f (ω 1 α )}. As before, define the catch-all set: And if this set is not empty, add its elements as singleton sets to create the κ-sized partition π f = {h 1 , . . . , h α , . . . } ∪ T 3 . Then, for each h ∈ π f , P(T 1 | h) = 0. In parallel fashion, with respect to function g, define π g so that for each h ∈ π g , P(T 2 |h) = 0. Since P(R) > 0, and by SA 5 at least one of T 1 and T 2 is not null, that is since maxi-mum{P(T 1 ), P(T 2 )} > 0, P is not conglomerable in at least one of these two partitions, π f and π g . ♦Lemma 3.5 The following example alerts the reader that Cases 1 and 2, where, respectively, ↑ and ↓ well order the set of tiers, are sufficiently dissimilar that for a countable state space Ω only one is consistent with P being countably additive. In the light of Example 3.6, the proof of nonconglomerability when ↓ is a well order (Case 2-Lemma 3.7) uses different reasoning than when ↑ is a well order (Case 1-Lemma 3.5), and shows that where P is conglomerable, it is concentrated on tiers with limit ordinal indices. This contradicts SA 5 , which requires that the union of points in tiers with successor ordinal indices have positive probability.
LEMMA 3.7. Suppose ↓ is a well order of the set of tiers, each of which is P-null-Case 2. Then P is nonconglomerable.
Consider the partition (a "histogram") of R according to the cardinality of each tier. That is, let π C = {h λ : where τ ∈ h λ if and only if |τ | = λ, and λ < κ}. In the light of Lemma 3.4, each tier has cardinality less than κ. So π C is a partition of the set of all tiers. That is, h 1 is the set of those tiers with exactly one point, {ω}; h n is the set of those tiers with exactly n-points, and for each cardinal λ < κ, h λ is the set of tiers each with exactly λ-many points. Since κ is regular and not weakly inaccessible, there are fewer than κ cardinals less than κ, |π C | < κ. By SA 6 , the cardinality of tiers is a measurable function. As |π C | < κ and P is λ -additive 2 for each cardinal λ < κ, by Lemma 1.10, h∈π C P(∪h) = P(R) = d > 0. Thus, there is at least one uncountable set of tiers, h * ∈ π C , such that P(∪h * ) > 0.
As h * is well ordered by ↓, according to SA 5 it can be partitioned into three disjoint measurable sets, where the first two (those tiers in h * with successor ordinal indices) are not both P-null.
(A) Is the set of successor tiers in h* each with an even ordinal index ending "+2n" for integer, n = 1, 2, . . . . (B) Is the set of successor tiers in h* each with an odd ordinal index ending "+2n-1" for integer, n = 1, 2, . . . . (C) the set of tiers in h* each with a limit ordinal as its index. For convenience, since 0 has no predecessor, we include the first element of h*, τ 0 , in C.
We construct two partitions. The first partition shows that if P is conglomerable, then P(∪A) = 0. The second partition shows that if P is conglomerable, then P(∪B) = 0. Together, this contradicts the final clause of SA 5 .
To create the first partition, pair each tier in the set A 1-1 with its immediate predecessor tier in h * . Since each tier in h * has a common cardinality, then pair, 1-1, each element of each tier in A with an element of its predecessor tier. Let f be this 1-1 pairing of points in ∪A with points in the ∪(predecessors-to-A). Write these pairs as {ω, f (ω)} where ω ∈ A ⊂ h * . Then, P({ω}|{ω, f (ω)}) = 0 for each such pair, since f is regressive on the ordinals indexing tiers in A. Complete the partition by adding all the singleton sets {ω} for ω ∈ ∪R − (∪A ∪ Range( f )) and denote an arbitrary element of this partition h B . Then, P(∪A|h B ) = 0, which gives us P(A) = 0 by conglomerability of P.
Similarly, to create the partition targeted at showing P(∪B) = 0, use a 1-1 regressive function to pair each element of the set of tiers B with its immediate predecessor tier in h * and continue the reasoning just as in the previous paragraph.
The upshot is that if P is conglomerable in each of these two partitions, we have a contradiction with SA 5 that requires that at least one of sets ∪A and ∪B is not P-null. ♦Lemma 3.7 REMARK. Lemma 3.7 is established by finding two, 1-1 regressive functions for the ordinals, respectively, indexing set A and indexing set B. But set C is stationary; hence, by Fodor's (1956) "Pressing Down" lemma, there is no such 1-1 regressive function on C. (See Jech (1978), p. 59.) We do not know whether, if P(∪C) > 0, P is nonconglomerable for a measurable event that is a subset of ∪C. That is, the elements of M ↓ satisfy τ m ↓ τ n and elements of N ↑ satisfy τ m ↑ τ n whenever n > m-Case 3. Then P is not conglomerable.
Proof. Combine the two sequences M ↓ and N ↑ to form a single countable set L, linearly ordered, either by ↑ or by ↓. The countably many tiers in the linear order L create a countable partition of all the tiers and, for convenience, consider the set R of all tiers other thanτ , and where P(R) > 0. Partition the linear order R by using the elements of L to form cuts, in the fashion of Dedekind Cuts. By SA 3 , these cuts produce measurable sets in R. Since each such interval is defined using no more than countably many elements of L, the intervals are measurable.
By Lemma 3.4, if P is conglomerable, and as it is countably additive, then one and only one of these countably many intervals is not null. Denote that interval I * 0 . That is, P(R) = P(∪I * 0 ). Thus P is a 0-1 distribution on these countably many intervals. Denote by I * ↑ 0 the interval of tiers above I * 0 , and by I * ↓ 0 the interval of tiers below I * 0 . By SA3, each of ∪ I * ↑ 0 and ∪ I * ↓ 0 is measurable. As P is σ -additive, P(∪ I * ↑ 0 ) = P(∪ I * ↓ 0 ) = 0. The linear order of tiers within the interval I * 0 is again one of the three types, corresponding to Cases 1, 2, or 3. If I * 0 produces a linear order that is a well order, corresponding to either Case 1 or 2, complete the argument by duplicating Lemma 3.5 or 3.7 (respectively) applied to the interval I * 0 . If the linear order within I * 0 is also an instance of Case 3, then repeat the reasoning to produce a subinterval, I * 1 ⊂ I * 0 , where P(R) = P(∪I * 1 ). We continue the argument, assuming that at each stage in the repetition of this reasoning the interval I * α has an internal linear structure corresponding to Case 3. Define the intervals I * α inductively. At successor ordinals β = α+1, create I * β by applying the reasoning, above, used to create I * 1 from I * 0 . At limit ordinals β ≤ κ, let I * β = ∩I * α for α < β. To see that these are measurable sets, define the two sequences of increasing "tail" intervals I * ↑ 0 ⊂ I * ↑ 1 ⊂ . . . and I * ↓ 0 ⊂ I * ↓ 1 ⊂ . . . . By SA 3 , for each α ≤ κ the sets ∪I * ↑ α and ∪I * ↓ α are measurable, being "Dedekind cuts" in the linear ordering of tiers. As ∪I * α = R -(∪I * ↑ α ∪ ∪I * ↓ α ), also ∪I * α is measurable. For each λ < κ, P is λ-additive 2 . So for each α < κ, P(∪I * ↑ α ) = P(∪I * ↓ α ) = 0. Therefore, for each α < κ, P(∪I * α ) = P(R).
Next, consider the two tail intervals formed by the cut at I * , I * ↑ , and I * ↓ , where I * ↓ is entirely below I * and I * is below I * ↑ in the linear order of tiers. There are two subcases to consider.
As an illustration of Sufficient Condition 1, use the uniform density function f to identify conditional probability given finite sets as uniform over those finite sets, as well. That is, when F = {ω 1 , . . . , ω k } is a finite subset of Ω with k-many points, let P(·|F) be the perfectly additive probability that is uniform on these k-many points. These conditional probabilities create a single tier τ = Ω, as P({ω 1 }|{ω 1 , ω 2 }) = 0.5 for each pair of points in Ω.
The usual theory of regular conditional distributions treats the example differently. We continue the example from that point of view. Consider the measure space <Ω, B, P> as above. Let the random variable X(ω) = ω, so that X has the uniform distribution on Ω. In order to consider conditional probability given the pair of points {ω, g(ω)}, let g(X) = (X/9) + 0.9 if 0 ≤ X < 0.9, = 9(X − 0.9) if 0.9 ≤ X < 1.
Let the sub-σ -sigma field A be generated by the random variable Y. The regular conditional distribution relative to this sub-σ -sigma field, P(B|A)(ω), is a real-valued function defined on Ω that is A-measurable and satisfies the integral equation In our case, then P[B|A](ω) almost surely satisfies P(X = 0.9Y|Y)(ω) = 0.9 and P(X = 0.1(Y + 9.0)|Y)(ω) = 0.1.
However, g(.09) = .91 = f (.09) and g(.91) = .09 = f (.91). That is, Y = 0.1 if and only if Z = 0.82. So in the received theory, it is permissible to have P(ω = .09|Y = 0.1) = 0.9 as evaluated with respect to the sub-σ -sigma field generated by Y, and also to have P(ω = .09|Z = 0.82}) = 0.5 as evaluated with respect to the sub-σ -sigma field generated by Z, even though the conditioning events are the same event. ♦Example 4.1 §5. Conclusion. Given a probability P that satisfies the six structural assumptions of the Proposition 3.2, we show that nonconglomerability of its coherent conditional probabilities is linked to the index of nonadditivity 2 of P. Specifically, assume P is not κ-additive 2 , and where κ is least and is not a weakly inaccessible cardinal. Then there is a κ-size partition π = {h α : α < κ} where the coherent conditional probabilities {P(· | h α )} are not conglomerable. Namely, there exists an event E and a real number ε > 0 where, for each h α ∈ π , P(E) > P(E | h α ) + ε.
The structural assumptions that we impose on the σ -field B reflect the constraint imposed by one part of Ulam's (1930) seminal finding, which applies when the state-space Ω is uncountable, | Ω| = κ ≥ ℵ 1 , when B includes each point in Ω, and P is σ -additive. If κ is not greater than a weakly inaccessible cardinal, then B cannot be the powerset of Ω. Because we do not want our findings to depend upon such a large cardinal assumption, we have to be cautious introducing measurable sets in our study about conglomerability in κ-sized partitions.
Without loss of generality, each countably additive probability can be completed by adding all subsets of each P-null set. So, we use P-complete countably additive measure spaces. As we explain, below, the six structural assumptions ensure that B is sufficiently rich for our study of nonconglomerability in large partitions, while being attentive to Ulam's Theorem that B cannot be as large as the powerset of Ω.
Our study takes the equivalence relation of a tier of points as the central concept, which is defined using conditional probability given finite sets of points: So singletons from Ω are required to be B-measurable (SA 1 ). Also, we require that tiers are measurable sets (SA 2 ). Since the tiers are linearly ordered and we consider sets of tiers above (and below) a given tier in this linear order, we require that intervals of tiers are measurable (SA 3 ). Taken together, SA 1 , SA 2 , and SA 3 make the linear order of tiers into a B-measurable function of the points in Ω. From this perspective, the last structural assumption, SA 6 , requires that the cardinality of tiers also is a B-measurable function. SA 4 and SA 5 are two "splitting" conditions. The former precludes such extreme σ -fields as when B is composed of countable/co-countable subsets of Ω, where binary (measurable) partitions of a non-null set are required to be of unequal cardinality. The second "splitting" condition SA 5 insures that when an uncountable set T of tiers is well ordered under the linear ordering of tiers, then the subset of tiers indexed with successor ordinals is not P-null if P(∪T) > 0, and that this subset of tiers can be further partitioned into two measurable subsets with the "odd" and "even" indices. This "splitting" ensures that when the linear order ↓ is a well order, we have measurable, regressive functions on tiers whose domain includes a non-null set.
The mutual consistency of these structural assumptions is evident for the simple case where |Ω| = κ = ℵ 1 adapted to Example 4.1, as follows.
EXAMPLE 5.1. Consider the P-complete measure space of Lebesgue measure on Lebesgue measurable subsets of the unit interval, under the Continuum Hypothesis. Then, as in Example 4.1,τ = ∅, assume a single nonempty tier, τ = Ω. SA 1 is satisfied, since the atoms of B are the singletons of Ω. SA 2 , SA 3 , SA 5 , and SA 6 are satisfied, trivially, because there is only one nonempty tier, Ω, which is measurable. SA 4 is satisfied since the unit interval contains an uncountable, (measurable) null set, e.g., the Cantor set. ♦Example 5.1 Next, with Example 5.2, we demonstrate that the five structural assumptions SA 1 , SA 2 , SA 3 , SA 5 , and SA 6 are jointly insufficient for the main Proposition. B is the smallest σ -field containing all singletons, i.e., B is the σ -field of countable/ co-countable subsets of Ω; and P({ω α }) = 0, for each α < ℵ 1 .
As in Example 4.1, assume there is a single tier. Hence, SA 1 is satisfied, since the atoms of B are the singletons of Ω. SA 2 , SA 3 , SA 5 , and SA 6 are satisfied, trivially, because there is only one nonempty tier, Ω, which is measurable. However, SA 4 is not satisfied, as each measurable binary partition of Ω produces sets of unequal cardinality.
Next, we establish that these conditional probabilities associated with the measure space <Ω, B, P> are conglomerable. If π is a countable partition of measurable events, then P is conglomerable in π as P is σ -additive. So, consider an uncountable partition of measurable events, π = {h α : h α ∈ B, α < ℵ 1 }. Note that if P fails to be conglomerable in π with respect to event E, then P fails to be conglomerable in π with respect to the complementary event E c . So, let E ∈ B with P(E) = 1. Then, for all but a denumerable set of elements of π , h α ⊆ E. Hence, by coherence, P(E | h α ) = 1 and P satisfies conglomerability in partition π , contrary to the conclusion of the Proposition 3.2. ♦Example 5.2 The Proposition 3.2 permits us to conclude that the anomalous phenomenon of nonconglomerability is a result of adopting the de Finetti/Dubins theory of coherent conditional probability instead of the rival Kolmogorovian theory of regular conditional distributions. It is not a result of the associated debate over whether probability is allowed to be merely finitely additive rather than satisfying countable additivity. Restated, our conclusion is that even when P is λ-additive 2 for each λ < κ, if P is not κ-additive 2 and has coherent conditional probabilities, then P will experience nonconglomerability in a κ-sized partition. The received theory of regular conditional distributions sidesteps nonconglomerability by allowing conditional probability to depend upon a subsigma field, rather than being defined given an event.