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Thread: Problems about probability

  1. #1
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    Problems about probability

    Let $\displaystyle (A_\beta)_{\beta \in B}$ be a family of pairwise disjoint events. Show that if $\displaystyle P(A_\beta) > 0 $ for each $\displaystyle \beta \in B$, then $\displaystyle B $ is at most countable.

    Anyone can help solve this problem? Thanks in advance.
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  2. #2
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    Quote Originally Posted by tszhin8831 View Post
    Let $\displaystyle (A_\beta)_{\beta \in B}$ be a family of pairwise disjoint events. Show that if $\displaystyle P(A_\beta) > 0 $ for each $\displaystyle \beta \in B$, then $\displaystyle B $ is at most countable.

    Anyone can help solve this problem? Thanks in advance.
    $\displaystyle B=\bigcup_{n=1}^{\infty}I_n,$ where $\displaystyle I_n= \left \{\beta \in B: \ \frac{1}{n+1} \leq P(A_{\beta}) \leq \frac{1}{n} \right \}.$ now if $\displaystyle |I_n|=\infty,$ for some $\displaystyle n,$ then $\displaystyle P(\bigcup_{\beta \in B}A_{\beta})=\infty,$ which is nonsense. so each $\displaystyle I_n$ is a finite set and thus $\displaystyle B$ is (at most) countable.
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