Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $m_{1}, \ldots, m_{n}$ be as in III.C.1. Deduce that the functions $m_{1}, \ldots, m_{n}$ are of class $C^{\infty}$.
Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $m_{1}, \ldots, m_{n}$ be as in III.C.1.\\
Deduce that the functions $m_{1}, \ldots, m_{n}$ are of class $C^{\infty}$.