By R.M. Blumenthal

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2) is again valid. For example, let EA be a metric space and 8, = W(E,) (the Bore1 sets). 2) holds. However if we regard (a, A, A , , XI)as taking values in (E,, 82)it will not, in general, be progressively measurable. When considering {9,} stopping times the following theorem allows us to stopping times in many situations. Let T be an { S t +stopping } time. Then for each p there such }that P'(T # T,,) = 0. exists a stopping time T,, relative to {9:+ Proof. 9) Then each T'") is an {F,} stopping time taking on the discrete set of values {k/2";k = 1,2, , ..

8) EX{f(Zs);r>= E"{Ns-,f(Z,); l-1. 8) is right continuous in s. 8) for all f E C implies its validity for all f E b6,. Thus we have established that {Z,, Y, ; t 2 0} is a Markov process over (W, Y, P") with transition function N for each x . Now suppose that g E bd',, f E C, t E [0, m), and {s,} is a sequence of rationals decreasing to t . ) g(X,))= E"{N,"-,f(X,) g(X,)), and letting n -, 00 we obtain EX{f(Z,dX,)) ) = E"{f(X,) g(X,)I. It now follows from MCT that E"{h(Z,, X,)}= E"{h(X,, X,)} for all h E b(6, x 6J, and this implies that P"{X, = Z,} = 1 (take, for example, h ( x , y ) to be a bounded metric for E J .

We will leave to the reader the task of developing the analogous (but simpler) situation in which T = (0, 1, . . , co}. 1) DEFINITION. provided { T I t} E 2FGrfor all t in [0, a). to {st,}) Note that {T = co} = {T < co}' E 9 = 9,, and ,so {T < t } E 9, for all t E T. Hence T E 9. Of course { T I f } E 9, if and only if {T > f } E S t . Clearly any nonnegative constant is a stopping time. Given the family {F,} we define for each t E [0, 00) a new a-algebra 9,+ = 9,. We have used the notation 9,+ for this a-algebra because it is the standard notation.