= Solution
{c}
Use the increasing nonnegative measurable functions $f_n(x)=x^{-2}\mathbf1_{[1/n,1]}(x)$. Their pointwise limit is the given function except at irrelevant endpoint choices, and the <fundamental theorem of calculus> on $[1/n,1]$ gives
$$
\int_{\mathbb R}f_n(x)\,dx=\int_{1/n}^1x^{-2}dx=n-1.
$$
<monotone convergence> therefore gives $\int f=\infty$. Since $f\geq0$, this is also $\int|f|$, and \b[$f$ is not <Lebesgue integrable>]. Measurability alone does not make a finite integral.
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