# Probabilities and Set Theory

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Let (Omega, A) be a measurable space, and P:A--> [0,infinity] an application such that P(AUB) = P(A) + P(B) when A,B E A and A intersection B = Ã¸, and P(Omega) = 1 . Prove that the following statements are equivalent:

(i) P is a probability

(ii) P is continuous on the decreasing series:

(iii) P is continuous on the increasing series:

(iv) and Ã¸

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Let be a mesurable space, and an application such that when and Ã¸, and . Prove that the following statements are equivalent:

(i) P is a probability

(ii) P is ...

#### Solution Summary

Proofs are provided for proabilities expressed with set theory. The solution is detailed and well presented.