Download e-book for kindle: Capacités et processus stochastiques by Claude Dellacherie

By Claude Dellacherie

ISBN-10: 0387056769

ISBN-13: 9780387056760

Dellacherie C. Capacites et processus stochastiques (fr)(ISBN 0387056769)(Springer, 1972)

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Example text

We use induction on m. Indeed, the statement is trivially true whenTO= 1. 10) that P*{Ef]Bm+1) = P*(BmDEn = p*(EnBm) Bm+1) + P*(Bg C\En Bm+1) + p*(EnAm+1) = J2n j:n

6. 5) for finite collections {Dn}. Suppose further that, whenever A\,A2,... n^00P(An) = 0. 5) for countable collections {Dn}. 7. Let J, P, P*, and M. 4). g. T = M, or T = CJ(J)). Let Q be any probability measure on T, such that Q(A) = P(A) for all AeJ. Then Q(A) = P*(A) for all AcT. Proof. For A 6 T, we compute P*(A) = infA 1 ,A 2 ,... ej Q(A) = Q = P on J U * Q(A), by countable subadditivity by monotonicity 20 2. PROBABILITY TRIPLES. e. P*(A) > Q(A). e. P*{A) < Q{A). Hence, P*{A) = Q(A). 8.

1 that these spaces present no difficulties. 6. COIN TOSSING AND OTHER MEASURES. 21 But suppose now that we wish to model the flipping of a (countably) infinite number of coins. ); rt = 0 or 1} be the collection of all binary sequences. But what about T and P? an = { ( r 1 , r 2 , . . ) G fi; rt = a* for 1 < i < n} . a„) = 1/2" for each set -A aia2 ... a„ ; n G N , a i , a 2 l . . A) for each A G J. 4), we need to verify that certain conditions are satisfied. 1. (a) Verify that the above J" is a semialgebra.

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Capacités et processus stochastiques by Claude Dellacherie

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