2.34

Exercise Let {\left|f_{n}\right|\le g\in L^{1}} and {f_{n}\rightarrow f} in measure. Show that,

  • (a) {\int f=\lim_{n\rightarrow\infty}\int f_{n}}.
  • (b) {f_{n}\rightarrow f} in {L^{1}}.

Proof: ~

  • (a) In Exercise 33 we proved a theorem analogous to Fatou’s lemma for {f_{n}\rightarrow f} in measure. We will use this analogue of Fatou’s lemma to prove an analogue of dominated convergence theorem. By taking real and imaginary parts, we can assume that {f_{n}} and {f} are real-valued. Applying Exercise 33 to {g+f_{n}} yields,

    \displaystyle  \int f\le\liminf\int f_{n}

    And applying Exercise 33 to {g-f_{n}} yields,

    \displaystyle  \limsup\int f_{n}\le\int f

    Therefore we have {\int f=\lim\int f_{n}}. Note that this almost exactly the same as the normal proof of dominated convergence theorem from Fatou’s lemma.

  • (b) It is easy to see that we have {\left|f_{n}-f\right|\rightarrow0} in measure and {\left|f_{n}-f\right|\le\left|f\right|+\left|f_{n}\right|\le2g} a.e. Therefore we apply (a) to get that {\lim_{n\rightarrow\infty}\int\left|f_{n}-f\right|=0} so {f_{n}\rightarrow f} in {L^{1}}.

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