I wonder about the following: let $f:[0,\infty]\to\mathbb{R}$ be Lebesgue integrable,i.e.
$\int_0^\infty |f(x)|d\lambda(x)<\infty$
Does it hold that $$\lim_{n\to\infty}\int_0^\infty |f_n(x)|d\lambda(x)=\int_0^\infty |f(x)|d\lambda(x)$$ if $f_n(x)=\sum_{k=0}^\infty f(kn^{-1})\mathbb{1}_{[kn^{-1},(k+1)n^{-1})}(x)$ ? If not, what are sufficient conditions?
As noted, obviously not, since changing $f$ on a set of measure zero can change the value of $\int|f_n|$. We fix that problem by assuming that $f$ is continuous. There are still counterexamples. But the counterexamples can't be totally trivial, because of the following positive result:
Suppose that $f:(0,\infty)\to(0,\infty)$ is continuous and non-increasing, and $\int_0^\infty f<\infty$. Then $$\frac1n\sum_{k=1}^\infty f\left(\frac kn\right)\to \int f.$$(We left out $k=0$ because $f(0)$ is undefined.)
Proof: $$\int_{1/n}^\infty f\le\frac1n\sum_{k=1}^\infty f\left(\frac kn\right)\le\int_0^\infty f,$$ and $$\int_0^{1/n}f\to0.$$