Riemann-Lebesgue lemma: Construction of weak convergent sequences from 1-periodic function

(1) Let $U:= \int_0^1 u(x)dx$. Since $u$ is one-periodic, $$ \frac1n \int_0^{nx} u(t)dt = \frac1n\left( \lfloor nx \rfloor \cdot U + r_n\right), $$ where $r_n$ is an integral of $u$ on a interval of length $\le1$: $$ r_n = \int_{\lfloor nx \rfloor}^{nx} u(s)ds \le \|u\|_{L^\infty}. $$ Since $$ nx-1\le \lfloor nx \rfloor \le nx, $$ it follows $\frac1n \lfloor nx \rfloor \to x$.

(2) Using the coordinate transform $(0,1)\mapsto (0,n)$ one can prove $\|f_n\|_{L^p}=\|u\|_{L^p}$. Similarly one can show $$ meas\{x: \ f_n(x)\ge c\} = meas\{x:\ u(x) \ge c\} $$ for all $c$. As strong convergence in $L^p$ implies convergence in measure, this should help to prove that $u=const$ in case the convergence is strong.