Exercise 11.10

Question

If $\mu(X) < +\infty$ and $f \in \mathscr{L}^2(\mu)$ on $X$, prove that $f
    \in \mathscr{L}(\mu)$ on $X$. If
    \begin{equation*}
      \mu(X) = +\infty,
    \end{equation*}
    this is false. For instance, if
    \begin{equation*}
      f(x) = \frac{1}{1+\lvert x vert},
    \end{equation*}
    then $f \in \mathscr{L}^2$ on $R^1$, but $f 
otin \mathscr{L}$ on $R^1$.

Amit Awasthi · Accepted Answer

\begin{proof}
      We have, by Theorem $11.35$,
      \begin{equation*}
        \int_X \lvert f \rvert\,d\mu \le \lVert f \rVert \cdot \lVert 1 \rVert =
        \lVert f \rVert \mu(X) < \infty.\qedhere
      \end{equation*}
    \end{proof}

Exercise 11.10

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Exercise 11.10

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