Let ( f n ) n ∈ N be a sequence of elements in M ( X , S ), let g ∈ M + ( X , S ) such that ∫gdμ&lt;∞ 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 μQuestion 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 μMy try: Define h n = f n + gThen 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 μSo we get ∫ E lim inf n → ∞ ( f n + g ) d μ ≤ lim inf n → ∞ ∫ E ( f n + g ) d μHow do we conclude?

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Let   (      f    n        )          n      ∈              N             be a sequence of elements in   M  (  X  ,  S  ), let   g  ∈      M    +    (  X  ,  S  ) such that ∫gdμ&amp;lt;∞ and       f    n    ≥  −  g      (  a  .  e  .  −  μ  )     ∀  n  ∈      N       in   E  ∈  S. Then, it follows from Fatou&#039;s lemma that:      ∫          E            lim inf          n      →      ∞        (      f    n    )  d  μ  ≤      lim inf          n      →      ∞            ∫    E        f    n    d  μQuestion 1 Can someone please give me a reference for the above generalised fatou&#039;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  μMy try: Define      h    n    =      f    n    +  gThen       h    n    ≥  0. Applying fatou&#039;s lemma to       h    n   we have      ∫          E            lim inf          n      →      ∞        (      h    n    )  d  μ  ≤      lim inf          n      →      ∞            ∫    E        h    n    d  μSo we get      ∫          E            lim inf          n      →      ∞        (      f    n    +  g  )  d  μ  ≤      lim inf          n      →      ∞            ∫    E    (      f    n    +  g  )  d  μHow do we conclude?

Perman7z · Accepted Answer

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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