A function $f:\mathbb{R} \rightarrow \mathbb{C}$ is called almost periodic if it is the uniform limit of trigonometric polynomials. One can show that for almost periodic $f$ the following pointwise limit (called the mean value of $f$) exists $$ M(f) = \lim_{x\rightarrow \infty} \frac{1}{x} \int_0^x f(t)dt.$$ We call a almost periodic function $f$ quasi-periodic if the $\mathbb{Z}$-module $$ \mathcal{M}(f)=\langle \nu\in \mathbb{R} : M(f\cdot e^{-i \nu x}) \neq 0 \rangle_{\mathbb{Z}} $$ (called the frequency module of $f$) is finitely generated. Further we set $$ \mathcal{A}(f)=\{ \text{ g quasi-periodic and }\mathcal{M}(g) \subseteq \mathcal{M}(f)\}.$$

My question is whether there are known results relating pointwise limit of quasi-periodic functions and the convergence of the associated mean values.

For example is there a version of Lebesgue's dominated convergence theorem for $M(\cdot)$? I think the minimal assumptions should look something like this

**Proposition:** Let $f$ be a quasi-periodic function and let $(f_n)_{n\in \mathbb{N}}\subseteq \mathcal{A}(f)$ be a sequence bounded in the supremum norm and $f_n \rightarrow f$ pointwise then $M(f_n)\rightarrow M(f)$.

Any reference would be greatly appreciated.

**Edit:** For me a trigonometric polynomial is a function of the form

$$ \sum_{j=1}^n a_n e^{i \nu_j x}, $$

where $n\in \mathbb{N}, a_j\in \mathbb{C}$ and $\nu_j\in \mathbb{R}$.