This question comes from Sheldon Axler's Measure, Integration, & Real Analysis textbook which I'm currently self-studying.
Problem 3A.12: Show that there exists a Borel measurable function $f: \mathbb{R} \to (0, \infty)$ such that $\int \mathcal{X}_I f \mathop{} \mathrm{d} \lambda = \infty$ for every nonempty open interval $I \subseteq R$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, $\mathcal{X}_I$ is the indicator function on $I$.
I'm have a bit of trouble coming up with a function that seems to make sense here.
One idea was to place "tall spikes" around each rational:
Let $q_1, q_2, \dots$ be an enumeration of the rationals. For each $q_k$, choose some small $\delta_k$ (maybe say $\leq 4^{-k}$) to construct $f_k(x) = \begin{cases} 4^k & x \in (q_k - \delta_k, q_k + \delta_k) \\ 0 & \text{otherwise}\end{cases}$ (we set the "height" of the spike such that the integral of $f_k$ is non-decreasing)
Let $f(x) := 1 + \sum_k f_k(x)$
Every subinterval of $f$ would have an unbounded integral and $f(x) > 0$ for all $x$. However, I'm not quite sure $f$ is well-defined since it seems like we could have values for $x$ that are unbounded, for example if $x$ is in infinitely many intervals $(q_k - \delta_k, q_k + \delta_k)$. Though I suppose that $\mu(\bigcup_{k \in \mathbb{N}^{+} } (q_k - \delta_k, q_k + \delta_k)) \leq \frac{2}{3}$, so maybe this isn't too common.
I'm not quite sure if I'm overlooking something dumb here.
