Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as:
Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 \in L^{\infty}(X,\mathscr{B},\mu)$, $$\frac{1}{N}\sum\limits_{n=1}^{N}U_T^{n}f_1U_T^{2n}f_2$$ converges in $L^2(X,\mathscr{B},\mu)$ (Here $U_T g:=g\circ T$). Moreover, for any $A\in \mathscr{B}$ with $\mu(A)>0$ we have $$\lim_{N\to \infty}\frac{1}{N}\sum\limits_{n=1}^{N} \mu(A\cap T^{-n}A \cap T^{-2n}A)>0.$$
Question: Can I deduce this theorem from the special case where $(X,\mathscr{B},\mu,T)$ is taken to be an invertible, ergodic, Borel probability system?
The reason I'm asking is that Einsiedler-Ward only seem to prove this for the special case. I'm not sure if i'm misreading their proof or if the reduction to the general case is easy.
My issue is primarily with the $L^2$ convergence claim. A first attempt at the reduction could be to apply the ergodic decomposition theorem. However this isn't a valid approach since our space isn't assumed to be a Borel space.