Let's say for simplicity that I'm on the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. In this setting, Riemann-Lebesgue lemma could be stated as: for any $f\in L^1(\mathbb{T})$, $$ \lim_{\vert k\vert\to\infty}\int_\mathbb{T} f(x)e^{ikx}\, dx =0$$ which can be interpreted as: for any $f\in L^1(\mathbb{T})$, $\widehat{f}\in c_0$, where $\widehat{\cdot}$ denotes the discrete Fourier transform and $c_0$ the space of infinitesimal sequences. If I considered a Radon measure $\mu$ on $\mathbb{T}$ instead of $f$, then it's easy to check that $\widehat{\mu}\in l^\infty$ and Riemann-Lebesgue lemma fails, take $\mu=\delta_x$. The case $d\mu = f\,dx$ with $f\in L^1$ corresponds to measures which are absolutely continuous w.r.t. the Lebesgue measure.
However the same technique of proof for $f\in L^1$, i.e. showing the statement for smooth functions and then exploiting a density argument, shows that Riemann-Lebesgue lemma holds for any $\mu$ belonging to the closure of smooth functions in the total variation norm $\Vert\cdot\Vert_{TV}$. So my question is: what is this space? Is it just the space of all absolutely continuous measures or it contains other objects (like the derivative of the Cantor function)? Does it have a proper characterization?
The class of measures for which the Riemann-Lebesgue lemma remains true is the well-known class of Rajchman measures See also This class is defined as follows $$ \mathcal R(\Bbb T)=\{\mu\in M(\Bbb T) :\widehat{\mu}(\xi)\to0, \quad|\xi|\to\infty\}.$$
Noet that identifying $L^1(\Bbb T)\equiv \{\mu=|f|d x: f\in L^1(\Bbb T)\}$ we have
$$L^1(\Bbb T)\subsetneq \mathcal R(\Bbb T)\subsetneq M(\Bbb T).$$ A characterization of Rajchmann's measure is given here