On Lexicographic Probability Relations

In this note I examine conditions under which a probability relation on a set of events is lexicographic. Chipman (1971) discussed the question of the minimal ordinal CT such that there exists an order-preserving function from an ordered set A (in his; paper A is fR with any order on it) to lRa with the lexicographic order on it. Since there always exists an ordinal /? such that there is an order-preserving function from A to the lexicographically ordered lRP (see Chipman), one can define an order as lexicographic if there is no order-preserving function from A to R. According to this definition, a countable set cannot be ordered lexicographically. Consider, however, the following example. Let Q be the set of all the finite unions of ra tional intervals [a, p) in [0,2). Q is closed under complementation and finite intersections. Define on Q a relation 2 as follows: for every A, BE Q, A Z B iff 11(~1n[O,l))>~(Bft[O,l)) or p(AfI[O,l))=p(BfI[O,l)) and p(An[l,2))> ~/t/In [l, 2)) (p denotes the Lebesgue measure). Q is countable, but seems to be lexicographic. Indeed, there exists no probability function P on Q such that A 2 B iff I’( 4) ZIP(B) (see Section 2). A natural definition of lexicographic orders seems therefore as follows. An order I? on a set X is lexicographic if there exist two orders R, and R2 on X such that for every x,y~ X, xRy iff xRly, but not yR,x; or xR,y, yR,x, and xR,y. (R, and R, may themselves be lexicographic orders.) This definition may encompass too much. For example, the decimal writing of real numbers induces on them an apparent lexicographic order. It seems, therefore, that if one wishes to avoid defining too simple orders as lexicographic, yet does not