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Math Help - continuity of probability distribution

  1. #1
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    continuity of probability distribution

    would you help me how I can prove this sentence? please.
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    Last edited by rough19; February 24th 2010 at 11:59 PM.
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  2. #2
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    Hello,

    For the last inequality P(\liminf_n A_n)\leq \liminf_n P(A_n) :

    Consider B_N=\bigcap_{n=N}^\infty A_n
    The sequence (B_N) is increasing.
    So P(\lim_N B_N)=\lim_N P(B_N) (basic property of a measure/probability)

    But \lim_N B_N=\liminf_n A_n

    So we have P(\liminf_n A_n)=\lim_N P(B_N)

    But \forall N,B_N \subseteq A_N. Thus P(B_N)\leq P(A_n)

    \Rightarrow \lim_N P(B_N)=\liminf_N P(B_N) \leq \liminf_n P(A_n)


    Finally, P(\liminf_n A_n)\leq \liminf P(A_n)


    (note : \liminf \equiv \underline{\lim})



    Does this help ?


    For the first inequality, consider taking the compliment of the last inequality.
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  3. #3
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    Absolutely.

    Thank you so much.
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