Integral limit for $L^2([0,1])$ functions

Question

Let $f \in L^2([0,1]).$

Prove or disprove that $\lim_{n \to \infty}\int_0^1(f(2^{-n}E(r2^n))-f(r))^2dr=0,$ where $E(\cdot)$ is the floor function.

I tried rewriting the integral: $\sum_{k=0}^{2^n-1}\int_{k2^{-n}}^{(k+1)2^{-n}}(f(k2^{-n})-f(r))^2dr.$ We also note that the result is correct for continuous functions using the dominated convergence theorem (or uniform continuity)

Any ideas how to prove the problem when $f \in L^2([0,1])$?

Answer 1

That's not true. Consider $f$ to be Dirichlet function ($1$ in rational points and $0$ in irrational). Then $f(2^{-n} E(r2^n)) = 1$, but $f(r)$ is zero a.e.