Consider the set of all trigonometrical polynomials of the form $P(x)=\sum_{j=1}^n a_n e^{ix\cdot\xi_n}$, where $\xi_n\in\mathbb{R}^d$. A function is said to be almost periodic in the sense of Bohr if it is a uniform limit of trigonometrical polynomials. Define mean value of a function $f\in L^1_{loc}(\mathbb{R}^d)$ to be the number $$\mathcal{M}(f)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T f(x)\,dx.$$ Then, a function $u$ is said to be almost periodic in the sense of Besicovitch if there is a sequence of trigonometrical polynomials $P_n$ such that $\mathcal{M}(|u-P_n|^2)\to 0$ as $n\to\infty$. In most texts, it is mentioned that the quantity $(\mathcal{M}(|u|^2))^{1/2}$ is a semi-norm on the set of all Besicovitch almost periodic functions, however I have not found any text which would provide an example of a non-zero Besicovitch almost periodic function $u$ with $\mathcal{M}(|u|^2)=0$.
Is anyone aware of such an example? I am not able to construct such an example and it is certainly not true for trigonometrical polynomials. Of course, I mean examples which are not zero on a set of positive measure.
Take any compactly suported $L^2$ function $f$ and set $P_n=0$, then $$ \lim_{n\rightarrow \infty} \mathcal{M}(\vert f - P_n \vert^2) = \mathcal{M}(\vert f \vert^2) =0 $$ thus, $f$ is Besicovitch almost periodic such that its seminorm vanishes.