Atomic decomposition of finite signed measures on compacts of R^n

Recently there has been interest in pairs of Banach spaces \((E_0,E)\) in an o-O relation and with \(E_0^{**}=E\). It is known that this can be done for Lipschitz spaces on suitable metric spaces. In this paper we consider the case of a compact subset \(K\) of \(\mathbf{R}^n\) with the Euclidean metric, which does not give an o-O structure, but we use part of the theory concerning these pairs to find an atomic decomposition of the predual of Lip\((K)\). In particular, since the space \(M(K)\) of finite signed measures on \(K\), when endowed with the Kantorovich-Rubinstein norm, has as dual space Lip\((K)\), we can give an atomic decomposition for this space