[分析] Hardy–Littlewood maximal function

[tsaihohan (有人想一起下圍棋嗎)]

        Given a Lebesgue measurable subset A of R^n, we

        denote by │A│ the Lebesgue measure of A. Given

        a Lebesgue integrable function g on R^n, the Hardy

        -Littlewood maximal function M_g of g is defined

        on R^n by


               M_g(x)=sup  ∫│g│ /│Br(x)│
                      r>0  Br(x)

        where Br(x) is the open ball with radius r centered

        at x屬於R^n.


        (a) Prove that for each t屬於R, the set {x屬於R^n:M_g(x)>t}

        is open.

        (b) It follows from (a) that M_g is a measurable function

        on R^n. Prove that when ∫│g│>0, we have ∫│M_g│=∞.
                                R^n                R^n

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