Exercise 7.6.13

Question

\item Show that if $f$ and $g$ are integralbe on $[a,b]$, then so is the product $fg$.
\item Show that if $g$ is integrable on $[a,b]$ and $f$ is continuous on the range of $g$, then the composition $f \circ g$ is integrable on $[a,b]$.

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item The set of discontinuities of $fg$ is the union of the set of discontinuities of $f$ and that of $g$. Since $f$ and $g$ are both integrable, these are both sets with measure zero, and hence their union is also measure zero; thus $fg$ is integrable.
\item Since $f$ is continuous, $f \circ g$ and $g$ share the same sets of discontinuity; therefore $g$ is integrable implies $f \circ g$ is integrable.
 \end{enumerate}

Exercise 7.6.13

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Exercise 7.6.13

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