Let µ be a nonnegative Radon measure on ℝd which satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ ℝd and r > 0, μ(B(x,r))C0rn, where B(x, r) is the open ball centered at x and having radius r . In this paper, when ℝd is not an initial cube which implies µ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paley g-function of Tolsa is bounded from the Hardy space H1 (µ) to L1(µ), and furthermore, that if f ∈ RBMO (µ), then [ġ(f )]2 is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)]2 belongs to RBLO (µ) with norm no more than CfRBMO(μ)2, where C0 is independent of f .