Exercise 1.8

Question

Put $f_n = \chi_E$ if $n$ is odd, $f_n = 1 - \chi_E$ if $n$ is even. What is the relevance of this example to Fatou's lemma?

Amit Awasthi · Accepted Answer

\begin{proof}
      Since we have
      \begin{multline*}
        \int_X \liminf_{n 	o \infty} f_n\,d\mu = 0\
        \lneq \min\{\mu(E),\mu(E^c)\} = 
        \liminf_{n 	o \infty}
        \int_X f_n\,d\mu
      \end{multline*}
      this example gives an example of when the inequality in Fatou's lemma is
      strict.
    \end{proof}

Exercise 1.8

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Exercise 1.8

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