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		paths. To help talk about these more interesting collections, the next definition describes what it means for a path in to converge and introduces some terminology about convergence of paths.

		9.20 Definition

		(a) We say that converges to j just in case, for all but finitely many i, .

		(b) We say that converges at node  r  just in case, for , and k is the least number such that for all , .

		(c) For each  r , C r  denotes .

		(d) We say that a path k-agrees with C r  just in case:

		(i) and ,

		(ii) , . . . , k, , and

		(iii) or .

		(e) For each  r  and each , C r ,k denotes .

		(f) For each , C(A) denotes .

		It follows from Lemma 9.11 that all of the C r 's, C r k's, and C(A)'s are measurable. We shall be interested in computing pr(C(A)) for particular sets A, and the C r 's and C r k,'s will be our avenue for doing this. The next lemma summarizes the relationship between the C(A)'s and C r 's and states some elementary properties of the C r 's and C r ,k,'s. We leave its proof to the reader.

		9.21 Lemma (Pitt [149, 150]) Fix  r  and k with . Then:

		(a) For each , .

		(b) For each  r  and and .

		(c) For each  r  C r ,k and .

		(d) For distinct  r  and  r ', .

		Fix an A and let . It follows from the above lemma that C(A) is the disjoint union of the members of C, and hence . The probability of C r , in turn, is the monotone decreasing limit of The next lemma shows how to compute the probabilities of the C r k's.