Let ( f n ) n ∈ N be a sequence of elements in M...

kreamykraka80

kreamykraka80

Answered

2022-07-10

Let ( f n ) n N be a sequence of elements in M ( X , S ), let g M + ( X , S ) such that ∫gdμ<∞ and f n g   ( a . e . μ )   n N   in E S. Then, it follows from Fatou's lemma that:
E lim inf n ( f n ) d μ lim inf n E f n d μ
Question 1 Can someone please give me a reference for the above generalised fatou's lemma.
Question 2 What is the condition on ( f n ) n N so that we may drop liminf and we have
E lim n ( f n ) d μ lim n E f n d μ
My try: Define
h n = f n + g
Then h n 0. Applying fatou's lemma to h n we have
E lim inf n ( h n ) d μ lim inf n E h n d μ
So we get
E lim inf n ( f n + g ) d μ lim inf n E ( f n + g ) d μ
How do we conclude?

Answer & Explanation

Perman7z

Perman7z

Expert

2022-07-11Added 13 answers

Let g n ( x ) = inf { f i ( x ) : i n }. Then g n ( x ) increases monotonically to lim inf f n and by the monotone convergence theorem, g n = lim inf f n . Since g n f n for every n, we have g n f n . Your result then follows.

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