Let ( X , A , μ ) be a measure space let f and f 1 , f 2 , … be nonnegative real-valued A -measurable functions on X that are μ-integrable.Suppose that they satisfyi) { f n } converges to f μ-a.e., andii) ∫ f d μ = lim n ∫ f n d μ.Show that lim n ∫ | f n − f | d ν = 0.I'm thinking I want to bound | f n − f | from above μ-a.e. by something μ-integrable, independent of n, and use the DCT, but I'm not successful yet. Any hints?

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Let   (  X  ,      A    ,  μ  ) be a measure space let f and       f    1    ,      f    2    ,  … be nonnegative real-valued       A  -measurable functions on X that are   μ-integrable.Suppose that they satisfyi)   {      f    n    } converges to f   μ-a.e., andii)   ∫  f  d  μ  =      lim    n    ∫      f    n    d  μ.Show that       lim    n    ∫  |      f    n    −  f  |  d  ν  =  0.I&#039;m thinking I want to bound   |      f    n    −  f  | from above   μ-a.e. by something   μ-integrable, independent of n, and use the DCT, but I&#039;m not successful yet. Any hints?

Blaine Foster · Accepted Answer

I tried to adapt the proof of the DCT:First notice that       |        f    n    −  f      |    ≤      |        f    n        |    +      |    f      |   for all n, then consider  ∫  2      |    f      |    =  ∫      lim inf    n        (          |              f      n              |        +          |        f          |        −          |              f      n        −    f          |        )    ≤      lim inf    n        (    ∫          |              f      n              |        +          |        f          |        −    ∫          |              f      n        −    f          |        )  and therefore (since   f  ,      f    n   are non-negative and integrals converge)  ∫  2  f  ≤      lim inf    n    ∫      f    n    +  f  −      lim sup    n    ∫      |        f    n    −  f      |    =  ∫  2  f  −      lim sup    n    ∫      |        f    n    −  f      |  and this implies      lim sup    n    ∫      |        f    n    −  f      |    ≤  0    ⟹        lim    n    ∫      |        f    n    −  f      |    =  0.

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