Let $\Omega \subset \mathbb R^d$ with $d \geq 2$ (I am mostly interested in the case when $\Omega$ is the unit ball). A vector field in $L^p(\Omega,\mathbb R^d)$ is called a *divergence measure field* if its distributional divergence is a Radon measure. The space of divergence measure fields over $\Omega$ is denoted by $\mathcal D \mathcal M^p(\Omega, \mathbb R^d)$.

My question seems trivial, but I couldn't find an answer anywhere.

**Question:** Let $\mathcal C(\Omega,\mathbb R^d)$ be the space of continuous vector fields over $\Omega$. Is it true that $\mathcal C(\Omega,\mathbb R^d) \subset \mathcal D \mathcal M^p(\Omega)$?

**Discussion:** The result clearly fails in one dimension, since then $\mathcal D \mathcal M^p(\Omega) = BV(\Omega) \cap L^p(\Omega)$ (with a ${}_{loc}$ if $\Omega$ is unbounded) and there exist continuous functions that are not in $BV$. However, in more than two dimensions the inclusion $BV(\Omega,\mathbb R^d) \cap L^p(\Omega,\mathbb R^d) \subset \mathcal D \mathcal M^p(\Omega,\mathbb R^d)$ is proper. For example,
$$
g(x,y) = \Big(\sin\big(\frac{1}{x-y}\big), \sin\big(\frac{1}{x-y}\big)\Big) \in \mathcal D\mathcal M^\infty(\mathbb R^2,\mathbb R^2) \setminus BV_{\mathrm{loc}}(\mathbb R^2,\mathbb R^2).
$$

There are quite a few papers on continuous solutions of
\begin{equation}\label{eq2}
\text{div}\, v = F, \tag{1}
\end{equation}
where $F$ is a distibution. The closest I came to some sort of an answer is Theorem 3.7 (see also Definitions 2.1 & 2.3) in *de Pauw and Pfeffer, Distributions for which div v = F has a continuous solution, 2007*. It says that a continuous solution of \eqref{eq2} exists if and only if for any compactly supported sequence of test functions $\varphi_i \in \mathcal D(\Omega, \mathcal R^d)$ it holds
\begin{equation}\label{eq3}
\lim_i |\varphi_i|_1 = 0 \quad \text{and} \quad \sup_i |\nabla \varphi_i|_1 < \infty \quad \implies \quad
\lim_i F(\varphi_i) = 0. \tag{2}
\end{equation}
A consequence would be that if there exists a distribution satisfying \eqref{eq3} which is not a Radon measure, then the inclusion I'm after would be false. However, I don't know if \eqref{eq3} implies that $F$ is a Radon measure (perhaps under additional assumptions on $\Omega$, such as compactness).

Any help will be appreciated.