# 實變問題(bounded variation)

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

thks

### 1 個解答

• 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