I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the literature they just write it off as an application of ergodic decomposition without much explanation (see for instance Corollary 3.16 in https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773063-8/S0002-9947-1985-0773063-8.pdf or even the original paper by Wiener and Wintner https://www.jstor.org/stable/2371534?seq=6) but I can't get it to work.
The most explicit reference I have found so far is Theorem 2.12 in Assani's book, where he uses ergodic decomposition to show that the function $$ F(x) = \sup_{\alpha\in \mathbb{T}}\limsup_{N,M \to \infty} \left\lvert \frac{1}{N}\sum_{n=1}^{N}f(T^nx)e(n\alpha) - \frac{1}{M}\sum_{m=1}^{M}f(T^mx)e(m\alpha)\right\rvert$$ is $0$ almost everywhere. However, it is is highly unclear why $F$ would even be measurable, so the argument feels iffy.
I think I can find a workaround by appealing to Bourgain's Uniform Wiener-Wintner, but thats definitely not the way Wiener and Wintner or the authors of the other paper I linked did it, as it was not yet available. The fact that they lightly mention to use ergodic decomposition makes me think that its either very simple, or at least they thought it was.
Updates/Clarifications
- I'm not wondering whether the Wiener-Wintner Ergodic Theorem is true or not for non-ergodic systems. For instance, one can prove a far reaching generalization of the result without assuming ergodicity using the Host-Kra machinery as in Theorem 14 in Chapter 23 of their book (https://www.ams.org/books/surv/236/). I'm wondering whether an ergodic decomposition argument works to lift the usual result to non-ergodic systems. Interestingly enough, in the preface of the proof of their generalization Host and Kra state: "...we do not assume ergodicity. Another approach would be to first prove the result for an ergodic system and then generalize this to arbitrary systems using the ergodic decomposition. However, there is a hidden difficulty with this approach, in that we do not know how to prove that the set of $x\in X$ such that the convergence holds for every nilsequence is measurable." So it seems that measurability issues in Wiener-Wintner have been addressed before.
- An alternative approach to that of Assani, could be something along the following lines. Letting $\mu = \int_{X}\mu_{x}\mu(x)$ be the ergodic decomposition of $\mu$, one can find a set $X_0\subseteq X$ with $\mu(X_0)=1$, such that for every $x\in X_0$ there exists a subset $X_x\subseteq X$ with $\mu_{x}(X_x)=1$, so that for every $y\in X_x$ the averages $$\frac{1}{N}\sum_{n=1}^{N}f(T^ny)e(n\alpha)$$ converge as $N\to \infty$ for every $\alpha\in \mathbb{T}$. Following the common intuition that says that ergodic decomposition sends each point an ergodic measure supported on an invariant component containing the point, one could now consider the set $$X' = \{x\in X_0 \colon x\in X_x\}.$$ If this set was measurable, then clearly $\mu(X')=1$, and we would be done. That said, showing that $X'$ is measurable has proven quite difficult aswell.
This has now been cross-posted to MO as suggested by John; see https://mathoverflow.net/q/503422
