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Math Help - Problems about probability

  1. #1
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    Problems about probability

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

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