Coherence of de Finetti coherence

Abstract

We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti’s notion of coherence.

Appendix: Background lemmas

In this section we collect all results needed for the proof of Theorem 2.1, together with detailed references where the interested reader can find a proof.

The following result has a central role in boolean algebra theory:

Lemma 2.1

(Stone representation theorem, (Koppelberg 1989, Chapter 3), (Sikorski 1960, §8)) For every boolean algebra D let $D'$ be the topological space given by the prime (=maximal) ideals of D equipped with the following (hull kernel, or Zariski) topology: a basic clopen of $D'$ is the set $\{\mathfrak m\in D'\mid d\in \mathfrak m\}$ of all maximal ideals to which an element $d\in D$ belongs, letting d range over all elements of D. An open set is an arbitrary union of these clopens. Then $D'$ turns out to be a totally disconnected compact Hausdorff space, called the Stone space of D. Let $\mathsf {CLOP}(D')$ denote the family of clopen subsets of $D'$. Then D is isomorphic to the boolean algebra $(\mathsf {CLOP}(D'),\emptyset , D',\cup ,\cap , \lnot ),$ where, for each $X\in \mathsf {CLOP}(D'),$ $\lnot X= D'\setminus X$ is the complement of X in $D'$.

Throughout this paper the boolean algebras A and B are identified with their isomorphic copies $(\mathsf {CLOP}(A'),\emptyset , A',\cup ,\cap , \lnot )$ and $(\mathsf {CLOP}(B'),\emptyset , B',\cup ,\cap , \lnot )$ given by Stone theorem.

Suppose now that $A\cup B$ generates C. Then [(Koppelberg 1989, p. 161), (Sikorski 1960, p. 40)], A and B are independent subalgebras of C iff the free product $A\oplus B$ (“boolean product” in (Sikorski 1960, p. 40) is isomorphic to C. The following lemma is a routine corollary of Stone duality:

Lemma 2.2

[(Koppelberg 1989, equation(1), p. 161], (Sikorski 1960, p. 41)) If A and B are independent subalgebras of C and $A\cup B$ generates C then C is isomorphic to the boolean algebra $(\mathsf {CLOP}(A'\times B'), \emptyset , A'\times B',\cup ,\cap , \lnot )$, where $A'\times B'$ is the cartesian product of the Stone spaces $A'$ and $B'$, with the product topology.

(Finitely additive, normalized) measures on a boolean algebra A have been defined in Subsect. 1.5.

Lemma 2.3

(De Finetti no Dutch Book theorem; de Finetti 1931, 1937, 1974; Dickey and Eaton 2009, Theorem 3.2; Kemeny 1955; Lehman 1955) Let A be a boolean algebra, $A^*$ a finite subset of A, and $\beta :A^*\rightarrow {{\mathrm{[0,1]}}}$ a map. Then $\beta$ is a coherent book iff it can be extended to a measure on A.

We refer to Tao (2011) and Tao (2010) for measure spaces, probability measures and $\sigma$-finiteness of measures and/or measure spaces.

Lemma 2.4

(Weak form of Carathéodory extension theorem; Tao 2010, pp. 277–280) Let S be a totally disconnected compact Hausdorff space, i.e., a Stone space. Then every finitely additive $\sigma$-finite measure m on the clopen subsets of S uniquely extends to a countably additive measure on the Borel sets of S. (If $m(S)=1$ then m is automatically $\sigma$-finite.)

Lemma 2.5

(Existence and uniqueness of product measure, Tao 2011, Proposition 1.7.11, p. 163) Suppose $(X,\mathcal B(X),\mu )$ and $(Y,\mathcal B(Y),\nu )$ are $\sigma$ -finite measure spaces. Then there is a unique measure $\xi$ on the product $\sigma$-algebra $\mathcal B(X)\times \mathcal B(Y)$ satisfying $\xi (E\times F)=\mu (E)\nu (F)$ for all $E\in \mathcal B(X)$ and $F\in \mathcal B(Y).$ $\xi$ is called the product measure of $\mu$ and $\nu$.

It would be desirable to have a direct proof of Theorem 2.1 that does not make use of all the lemmas of this section.