Exercise 7.4.3

Question

Decide which of the following conjectures is true and supply a short proof. For those that are not true, give a counterexample.
\begin{enumerate}[label=(\alph*)] 
\item If $|f|$ is integrable on $[a, b]$, then $f$ is also integrable on this set.
\item Assume $g$ is integrable and $g(x) \geq 0$ on $[a, b]$. If $g(x)>0$ for an infinite number of points $x \in[a, b]$, then $\int_{a}^{b} g>0$.
\item If $g$ is continuous on $[a, b]$ and $g(x) \geq 0$ with $g(y_0)>0$ for at least one point $y_{0} \in[a, b]$, then $\int_{a}^{b} g>0$.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Letting $d$ be Dirichlet's function, let $f(x) = d(x) - 1/2$ (not integrable), with $|f| = 1/2$ (integrable)
\item $h(x)$ in Exercise 7.3.9 is a counterexample
\item Let $\epsilon = g(y_0) /2 $. Since $g$ is continuous there must be some $\delta> 0$ so that over $V_\delta(y_0)$,$g > y_0 / 2$. Then let
$$h(x) = \begin{cases}
    g(y_0) / 2 & x \in V_\delta(y_0) \
    0 & \text{otherwise}
\end{cases}$$
then since $g \geq h$, $\int^b_a g \geq \int^b_a h = \frac{\delta g(y_0)}{2} > 0$.
 \end{enumerate}

Exercise 7.4.3

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Exercise 7.4.3

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