幼稚園小孩 發問時間: 科學數學 · 1 0 年前

實變問題(bounded variation)

prove that if g:[a,b]->R is a function of bounded variation,then g is Lebesgue measurable.

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1 個解答

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  • 1 0 年前
    最佳解答

    我先證: f is continuous almost everywhere, then f is measurable

    Pf : Let E={x in [a,b], f is continuous at x}, F=[a,b]\E

    then [a,b]=E∪F

    F has measure zero so is measurable

    and E=[a,b]\F is also measurable(兩個可測集的交集也是可測)

    so f is measurable

    回到原問題 g is a function of bounded variation then g=g1-g2

    g1 and g2 are monone increasing and so there are measure zero sets E,F such that

    g1 (x) is continuous for all x in [a,b]\E, g2(y) is continuous for all y in [a,b]\F

    取A=E∪F, then A has measure zero and g1-g2 is continuous on [a,b]\A

    即 g is continuous a.e on [a,b] and so is measurable

    參考資料: 以前所學
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