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Math Help - Holder's inequality

  1. #1
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    Holder's inequality

    With s>1 and r>s. How can I show that L^s \supset L^r, for expectation with respect to a probability measure?
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    Hello,

    Show what you've done. Basically, you just have to apply Holder's inequality to E[X^s], with X^r as f and 1 as g. If X is in L^r then E[X^r]<\infty, that's how it'll work.
    Give it a try.
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  3. #3
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    Holder inequality: E|XY|\le (E|X|^p)^{1/p}(E|Y|^q)^{1/q}
    Let q=r/s
    with r.v. |Y|^s \text{ and } 1_\Omega
    Then E|Y|^s\le (E|Y|^r)^{s/r}<\infty?

    Not very sure though
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  4. #4
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    This is exactly it !

    Now you have to talk a little :

    Let's take Y\in L^r. Then E[|Y|^r]<\infty. - your calculations - . Thus Y\in L^s, which implies that L^r\subset L^s : any element of the first belongs to the second.
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