Exercise 11.4

Question

Exercise 4: If $f\in\mathscr L(\mu)$ on $E$ and $g$ is bounded and measurable on $E$, then $fg\in\mathscr L(\mu)$ on $E$.

analambanomenos · Accepted Answer

By Theorem 11.18 $fg$ is measurable, and if $|g(x)|<C$ for some real number $C$, then
    $$
        \int_E|fg|\,d(\mu)\le C\int_E|f|\,d(\mu)<\infty
    $$
    so that $fg\in\mathscr L(\mu)$.

Amit Awasthi · Answer

\begin{proof}
      If $\lvert g(x) \rvert \le M$ for all $x$,
      then $\int_E \lvert fg \rvert\,d\mu \le M\int_E \lvert
      f \rvert\,d\mu < \infty$ so $fg \in \mathscr{L}(\mu)$.
\end{proof}

Exercise 11.4

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Exercise 11.4

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