Mathema Tics: M. Marden

terminating graphs r(n) which contain at least one particle in generation n, but none in the next. By an interval of order n is meant the set i(r(n)) of all graphs -y such that tn = Tn where r(nf) is a particular graph in T.. The only interval of order 0 is ri itself. We now define interval to be either 0, or any single graph y, terminating or not, or any interval of order n, n = 0, 1,. It is easy to see that axioms I.1, 2, 3 are satisfied. Indeed we have the simple property that the intersection ij of two intervals is either 0 or i or j. 4. Measure in the Space ofGraphs.-While the notions of graph, distance and interval are geometric in character and depend only on the number t of types, measure may be introduced in various ways. One of the simplest, but by no means the only one to which the theory has been applied, may be imposed as follows. Suppose we assign a probability p(i; ji, . .. , it) to the event that a particle of type i should produce, upon transformation, jB + . . . + jt particles, j, of type v. Then every segment Yn of a graph -y has an associated probability p(-y,) that the event described by yn should occur. If we assign to intervals a measure by m(0) = 0 m(r(8)) = p(rl), m(i(r(ff))) = p(Tnn'), and m(y) = lim p('y,) for 'y nonterminating it turns out that M.1, 2, 3 are satisfied and thus Borel sets based on our intervals are measurable in the sense of §2. (Non-compactness of intervals makes M.3 non-trivial.) The procedure applies in much more geteral systems where the transition probabilities are functions of the time.