Given a function \(H\in L^ 1 (\mathbb{R}^ n)\) a measurable set \(E\subset \mathbb{R}^ n\) is said to have variational mean curvature \(H\) if \(E\) minimizes the functional \(F_ H (E)= \int| D\chi_ E|+ \int_ E H(x)dx\), where \(\int| D\chi_ E|\) denotes the total variation of the vector measure \(D\chi_ E\), \(\chi_ E=\) characteristic function of the set \(E\). Conversely, it was shown by \textit{E. Barozzi} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. IX. Ser. Rend. Lincei, Mat. Appl. 5, No. 2, 149-159 (1994; Zbl 0809.49038)] that every subset \(E\) of \(\mathbb{R}^ n\) with \(\int| D \chi_ e| 1\). In further sections the authors investigate the regularity properties of Caccioppoli sets with mean curvature in \(L^ p\) for \(p\geq n\). Assuming \(p>n\) the reduced boundary \(\partial^* E\) is a smooth \((n-1)\)-dimensional manifold and \({\mathcal H}^ s (\partial E- \partial^* E)=0\) for \(s>n -8\). For \(p=n\) the just mentioned strong regularity theorem fails to hold which is shown by a counterexample. One can prove the following result: if \(0\in \partial E\) and \(E\) has mean curvature \(H\) in \(L^ p\) with \(p\geq n\), then the sets \(\lambda E= \{\lambda x\): \(x\in E\}\) converge in measure as \(\lambda\to \infty\) to a minimal cone.