Let { f n } be a sequence of measurable functions on a domain E and p be a positive finite real number such that(a) { f n } converges to a measurable function f almost everywhere, and;(b) lim n → ∞ ( | | f n | | p − | | f | | p ) = 0.Prove that | | f n − f | | p → 0.I believe what we have to show is that lim n → ∞ ∫ E | f n − f | p = 0.My approach is to hope to invoke the Dominated Convergence theorem, i.e., prove that there exists some g ∈ L 1 ( E ) such that | f n | ≤ g for almost everywhere for all n, then we can have ∫ E | f n − f | → 0.Then by (a), we can find an integer K such that | f k − f | &lt; 1for all k ≥ K. Then 0 ≤ ∫ E | f k − f | p ≤ ∫ E | f k − f | .Since the integral on the right approaches 0, we conclude that the same is true for the middle term and we will be done.However, I have no idea how to obtain the g as mentioned before, as well as use condition (b) given in the question. Some help would be appreciated.

Question

Let   {      f    n    } be a sequence of measurable functions on a domain   E and p be a positive finite real number such that(a)   {      f    n    } converges to a measurable function   f almost everywhere, and;(b)       lim          n      →      ∞        (      |        |        f    n        |              |        p    −      |        |    f      |              |        p    )  =  0.Prove that       |        |        f    n    −  f      |              |        p    →  0.I believe what we have to show is that      lim          n      →      ∞            ∫    E        |        f    n    −  f            |        p    =  0.My approach is to hope to invoke the Dominated Convergence theorem, i.e., prove that there exists some   g  ∈      L    1    (  E  ) such that       |        f    n        |    ≤  g for almost everywhere for all   n, then we can have      ∫    E        |        f    n    −  f      |    →  0.Then by (a), we can find an integer   K such that      |        f    k    −  f      |    &amp;lt;  1for all   k  ≥  K. Then  0  ≤      ∫    E        |        f    k    −  f            |        p    ≤      ∫    E        |        f    k    −  f      |    .Since the integral on the right approaches 0, we conclude that the same is true for the middle term and we will be done.However, I have no idea how to obtain the   g as mentioned before, as well as use condition (b) given in the question. Some help would be appreciated.

sleuteleni7 · Accepted Answer

This question has been answered many many times for   p  ≥  1. For   0  &amp;lt;  p  &amp;lt;  1 this follows from Fatou&#039;s Lemma:       |    f  −      f    n              |              p        ≤      |    f            |              p        +      |        f    n              |              p       in this case so       |    f            |              p        +      |        f    n              |              p        −      |    f  −      f    n              |              p       is non-negative Fatou&#039;s Lemma applied to this sequence gives the result immediately.

Let <mo fence="false" stretchy="false">{ f n </msub> <mo fence="false" stret

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