Let $f$ be a bounded measurable function on $\mathbb{R}$. We may consider its derivative $f'$ as a distribution on $\mathbb{R}$.
Is there a reasonable description of those distributions $\psi$ which arise in this way, i.e. are of the form $\psi = f'$ for some bounded measurable $f$?
For instance, $f'$ need not be a measure (it is a measure iff $f$ has bounded variation). On the other hand, the doublet $\psi = \delta_0'$ is not of this form.
Considering $f \in L^1_{\mathrm{loc}}$ would also be interesting.
More generally, on $\mathbb{R}^n$, I would also like to know which distributions are of the form $\psi = \operatorname{div} F$ for $F : \mathbb{R}^n \to \mathbb{R}^n$ bounded and measurable (or $L^1_{\mathrm{loc}}$).