We have $(X_i)_{i \in \mathbb Z}$ iid random variables with $1\le X_i \le2$ almost surely.
We define $X(x,\omega) \equiv X_i (\omega)$ if $x\in [i,i+1[$ (so it is bounded almost surely) and, for $\epsilon >0$, $X_\epsilon (x, \omega) \equiv X(x/\epsilon, \omega)$.
Define for $x\in [0,1]$
$$err_\epsilon(x,\omega)=\frac {\int_0^1 \frac {F(y)}{X_\epsilon(y,\omega)} dy}{X_\epsilon(x,\omega)\int_0^1 \frac {1}{X_\epsilon(y,\omega)} dy}$$
where $F$ is an $L^1(]0,1[)$ function (we can add continuity if it helps).
We would like to find functions $g$ and $f$ such that
$$\Vert err_\epsilon - g\Vert_{L^2(]0,1[ \times \Omega)} \le f(\epsilon) \to 0$$
when $\epsilon \to 0^+$
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