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Math Help - probability measure

  1. #1
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    probability measure

    Let P a probability measure and R a random variable on a certain space.
    P(.) only takes on the values 0 and 1 for all events, I have to show that there is a number a, such that the event T=a, has probability one.

    My idea was that P(.) can not be zero for all values, because the sum of P(.) must be 1. How do I work this out rigorously?
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  2. #2
    Super Member girdav's Avatar
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    Re: probability measure

    Let f:x\mapsto P(R\leq x). Let S=\left\{x\in\mathbb R: f(x)=1\right\}. By the hypothesis, the set S is non-empty. What about \inf S?
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