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Math Help - Prove limit inferior (lim inf) is measurable?

  1. #1
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    Prove limit inferior (lim inf) is measurable?

    f_n measurable

    Prove lim inf f_n is measurable?

    There's supposedly a proof in Royden's Real Analysis, but unfortunately I do not have access to it.

    Anyone?

    Much appreciated.
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  2. #2
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    for all a in R, {x in E: limsup{f_n: n in N}(x)>a},
    ={x in E: limsup{f_n(x): n in N}>a}
    = Union {x in E: f_n(x) > a}
    = a measurable set
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  3. #3
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    Quote Originally Posted by Idealconvergence View Post
    for all a in R, {x in E: limsup{f_n: n in N}(x)>a},
    ={x in E: limsup{f_n(x): n in N}>a}
    = Union {x in E: f_n(x) > a}
    = a measurable set
    Thank you. But that's for lim sup (I already knew that proof). For lim inf I can't just take the intersection as it messes up difference between >= and >. I think I have to go about by investigating the complements but I'm not sure.
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  4. #4
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    omg.

    lim inf f_n = -lim sup (-f_n)

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