Suppose that I have a set of null (Lebesgue) measure $N\subset \mathbb{R}$ such that $$\forall t\in \mathbb{R}\setminus N,\qquad \lim_{p\to + \infty} \sum_{i=1}^p b_n f_n(t) \text{ exists and belongs to } \mathbb{R},$$ where $b_n \in \mathbb{R}$ and $f_n$ are in $\mathcal{C}^\infty(\mathbb{R})$ and $T$-periodic functions. We define a function $f$ such that for $t\in \mathbb{R}\setminus N$, $$f(t) = \sum_{i=1}^{+\infty} b_n f_n(t).$$
Now take a $T$-periodic function $g\in L^2_{\text{loc}}$ such that $g= f$ almost everywhere in $\mathbb{R}$. Setting $G = g(T \cdot) \in L^2(]0,1[)$, do we have $$\left(\left( g= f\text{ almost everywhere in }\mathbb{R} \right) \qquad \Leftrightarrow \qquad G = \sum_{i=1}^{+\infty} b_n f_n(T \cdot) \text{ in } L^2(]0,1[) \right) ? $$
For me, we have $$ \begin{array}{cl} & \left( g= f\text{ almost everywhere in }\mathbb{R} \right) \\ \Leftrightarrow & \left( g= f\text{ almost everywhere in } [0,T] \right)\\ \Leftrightarrow & \left( G= \sum_{i=1}^{+\infty} b_n f_n(T \cdot)\text{ almost everywhere in }[0,1] \right) \\ \Leftrightarrow & \left( G = \sum_{i=1}^{+\infty} b_n f_n(T \cdot) \text{ in } L^2(]0,1[) \right) \\ \end{array} $$ But I might be missing something. In particular, is it true that if $G = \sum_{i=1}^{+\infty} b_n f_n(T \cdot)$ in $L^2(]0,1[)$ then there exits a set of null (Lebesgue) measure $N\subset \mathbb{R}$ such that $$\forall t\in \mathbb{R}\setminus N,\qquad \lim_{p\to + \infty} \sum_{i=1}^p b_n f_n(T t) \text{ exists and belongs to } \mathbb{R} ?$$