$\newcommand{\R}{\Bbb R}$ Consider the Lebesgue measure in $\R$ and the following proposition:
P. For each representative of a function class $f\in L^2[0,1]$ there is a sequence of continuous functions $(f_n)_{n\in\Bbb N}$ such that:
- $|f_n-f|$ is Riemann integrable on $[0,1]$, for all $n\in\Bbb N$.
- $\lim\limits_{n\to\infty} \int\limits_0^1 |f_n(x)-f(x)|^2\ \mathrm d x=0$.
There is no reason why this proposition should be true, but I cannot find a counterexample.