Letμ be a nonnegative Radon measure on Rd which only satisfies μ(B(x,r)) ≤ Corn for all x ∈ Rd,r ＞ 0,and some fixed constants Co ＞ 0 and n ∈ (0,d].In this paper,some weighted weak type estimates with Aρp,(log L)σ (μ) weights are established for the commutators generated by Calderón-Zygmund singular integral operators with RBMO(μ) functions.