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dottidot
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Independence Problem: Please Help!
I have been trying to figure out the proof to this problem for the past couple of days and still don't have an answer. The question is as follows:
Let (Q,F,P) be a probability triple such that Q is countable. Prove that it is impossible for there to exist a sequence A1,A2,A3,... E F which is independent, such that P(Ai) = 1/2 for eash i. [Hint: First prove that for each w E Q, and each n E N, we have P({w}) <= 1/(2^n). Then derive a contradiction.
Note that Q, represents Omega, F the sigma-algebra/sigma-fiels and E is "an element of or member of"
Would appreciate any help with this.
I have been trying to figure out the proof to this problem for the past couple of days and still don't have an answer. The question is as follows:
Let (Q,F,P) be a probability triple such that Q is countable. Prove that it is impossible for there to exist a sequence A1,A2,A3,... E F which is independent, such that P(Ai) = 1/2 for eash i. [Hint: First prove that for each w E Q, and each n E N, we have P({w}) <= 1/(2^n). Then derive a contradiction.
Note that Q, represents Omega, F the sigma-algebra/sigma-fiels and E is "an element of or member of"
Would appreciate any help with this.