Exercise 1.13

Question

Show that proposition $1.24(c)$ is also true when $c = \infty$.

Amit Awasthi · Accepted Answer

\begin{proof}
      We want to show that if $f \ge 0$ and $c = \infty$,
      \begin{equation*}
        \int_E cf\,d\mu = c\int_E f\,d\mu.
      \end{equation*}
      We can assume without loss of generality that $f > 0$ on $E$ by
      restricting to $E$, since $0 \cdot \infty = 0$ by definition $(1.22)$.
      Then, the integral of both sides is infinite, so we are done.
    \end{proof}

Exercise 1.13

Answers

Comments

Exercise 1.13

Answers

Comments

Add answer