Let $\mu $ be a nonnegative Radon measure on ${{\mathbb R}^d}$ which only satisfies $\mu (B(x, r))\le C_0r^n$ for all $x\in {{\mathbb R}^d}$, $r>0$, with some fixed constants $C_0>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathop{\rm RBMO}(\mu )$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.