實變數函數論

Question

Let {fk} be a sequence of nonnegative measurable function defined on EIf fk->f and fk≦f a.e on E,show that ∫Efk->∫Ef

Answer 1

是在實數還是廣義實數系統下看?

2006-11-30 00:49:01 補充：
f_k ≦ f a.e. => ∫_E f_k ≦ ∫_E f for all kSo limsup(k->oo) ∫_E f_k ≦ ∫_E f ... (1)Fatou lemma => ∫_E f ≦ liminf(k->oo) ∫_E f_k ... (2)(1) together with (2), the reslut follows.

Answer 2

它的題目就是這樣

2006-11-30 00:31:02 補充：
提示：利用Fatou's Lemma