Consider a non-atomic probability space $(X,\mathcal{B}, m)$. Let $T: X \to X $ be an ergodic invertible measure preserving transformation. Let $U_T$ be the Koopman operator associated with $T$. Show that every point of the unit circle $K$ is an approximate eigenvalue of $U_T$.
The hint was to use Rokhlin's lemma. But I am clueless about constructing the sequence of $L^2$ functions $f_n$ from Rokhlin's lemma which would help me prove that every point of $K$ is an approximate eigenvalue.
Let $\lambda\in \mathbb{S}^1=K$ be fixed. For each $\epsilon>0$ and $n\in\mathbb{N}$, find a set $E_{n,\epsilon}$ such that $T^iE_{n,\epsilon}$, $0\leq i\leq n$, are all disjoint and their union has measure $\geq 1-\epsilon$ (in particular $E_{n,\epsilon}$ has measure $\leq\frac{1}{n+1}$).
Define $f_{n,\epsilon}$ by putting $f_{n,\epsilon}(T^i x)=\lambda^i$ for $x\in E_{n,\epsilon}$. This defines $f_{n,\epsilon}$ on $\bigcup_{i=0}^n T^iE_{n,\epsilon}$. Define $f_{n,\epsilon}=1$ on any remaining points, so that $\Vert f_{n,\epsilon}\Vert_2=1$.
Now note that if $0\leq i\leq n-1$ and $x\in E_{n,\epsilon}$, we have $$U_T(f_{n,\epsilon})(T^ix)=f_{n,\epsilon}(T^{i+1}x)=\lambda^{i+1}=\lambda f_{n,\epsilon}(T^i x).$$ This show that $U_Tf_{n,\epsilon}=\lambda f_{n,\epsilon}$ on $\bigcup_{i=0}^{n-1} T^iE_{n,\epsilon}$, which has measure $\geq 1-\epsilon-\frac{1}{n+1}$.
So $\Vert U_T f_{n,\epsilon}-\lambda f_{n,\epsilon}\Vert_2\leq 2^{1/2}\left(\epsilon+\frac{1}{n+1}\right)^{1/2}$. The sequence $f_{n,n^{-1}}$ satisfies the condition for $\lambda$ being in the approximate spectrum.