Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hormander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1,∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1,∞).