Exercise 1.12

Question

Suppose $f \in L^1(\mu)$. Prove that to each $\epsilon > 0$ there exists a $\delta > 0$ such that $\int_E\lvert f \rvert\,d\mu < \epsilon$ whenever $\mu(E) < \delta$.

Amit Awasthi · Accepted Answer

\begin{proof}
      Let $E_N \coloneqq \{x \in X \mid \lvert f(x) vert \le N\}$. Then,
      letting $f_N \coloneqq \lvert f vert \chi_{E_N}$, we have
      $\lvert f_{N} vert \le f_{N+1}$, and the $f_N 	o \lvert f vert$,
      so by the monotone convergence
      theorem, for every $\epsilon > 0$ there exists an $N$ such that
      \begin{equation*}
        \int_X (\lvert f vert - f_n) \,d\mu < \frac{\epsilon}{2}
      \end{equation*}
      for all $n \ge N$. Now let $\delta$ such that $N\delta < \epsilon/2$.
      Then,
      \begin{align*}
        \int_E \lvert f vert\,d\mu &\le \int_E (\lvert f vert - f_n)\,d\mu +
        \int_E f_n\,d\mu\
        &\le \int_X (\lvert f vert - f_n)\,d\mu + N\delta < \epsilon.\qedhere
      \end{align*}
    \end{proof}

Exercise 1.12

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Exercise 1.12

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