實分析(證明題)有關於非負的可積simple函數

Question

If f is a nonnegative integrable simple function and ∫fdu=0,then f=0 a.e.

Answer 1

最佳解答

先證一個性質: f is nonegative measurable function and ∫_Afdu=0 for every measurable set , then f=0 a.e

proof Let E={x: f(x)>0}, E_n={x: f(x)>1/n)} then E=∪E_n 且 E_n is 是遞增的可測集合列

故(1/n)m(E_n)=∫_E_n 1/ndu<∫_E_n fdu<=∫fdu=0 for all n

=>m(E_n)=0 for all n

=>m(E)=0

所以 f=0 a.e

because f is nonegative integrable simple function for every measurable set F

0<=∫_Ffdu<=∫fdu=0

=>∫_Ffdu=0

故由以上的性質 f=0 a.e

參考資料： 以前所學